Simple Numerical Schemes for the Korteweg-deVries Equation
C. J. McKinstrie; M. V. Kozlov
2000-12-01
Two numerical schemes, which simulate the propagation of dispersive non-linear waves, are described. The first is a split-step Fourier scheme for the Korteweg-de Vries (KdV) equation. The second is a finite-difference scheme for the modified KdV equation. The stability and accuracy of both schemes are discussed. These simple schemes can be used to study a wide variety of physical processes that involve dispersive nonlinear waves.
Multiple soliton production and the Korteweg-de Vries equation.
NASA Technical Reports Server (NTRS)
Hershkowitz, N.; Romesser, T.; Montgomery, D.
1972-01-01
Compressive square-wave pulses are launched in a double-plasma device. Their evolution is interpreted according to the Korteweg-de Vries description of Washimi and Taniuti. Square-wave pulses are an excitation for which an explicit solution of the Schrodinger equation permits an analytical prediction of the number and amplitude of emergent solitons. Bursts of energetic particles (pseudowaves) appear above excitation voltages greater than an electron thermal energy, and may be mistaken for solitons.
Group-theoretical interpretation of the Korteweg-de Vries type equations
NASA Astrophysics Data System (ADS)
Perelomov, A. M.
1981-07-01
The Korteweg-de Vries equation is studied in the frame of the group-theoretical approach. Analogous equations have been obtained for which the multi-dimensional Schrödinger equation (with nonlocal potential) is of the same importance as the one-dimensional Schrödinger equation in the theory of the Korteweg-de Vries equation.
Group-theoretical interpretation of the Korteweg-de Vries type equations
NASA Astrophysics Data System (ADS)
Berezin, F. A.; Perelomov, A. M.
1980-06-01
The Korteweg-de Vries equation is studied within the group-theoretical framework. Analogous equations are obtained for which the many-dimensional Schrödinger equation (with nonlocal potential) plays the same role as the one-dimensional Schrödinger equation does in the theory of the Korteweg-de Vries equation.
The transformations between N = 2 supersymmetric Korteweg-de Vries and Harry Dym equations
NASA Astrophysics Data System (ADS)
Tian, Kai; Liu, Q. P.
2012-05-01
The N = 2 supercomformal transformations are employed to study supersymmetric integrable systems. It is proved that two known N = 2 supersymmetric Harry Dym equations are transformed into two N = 2 supersymmetric modified Korteweg-de Vries equations, thus are connected with two N = 2 supersymmetric Korteweg-de Vries equations.
The Korteweg-de Vries equation: its place in the development of nonlinear physics
NASA Astrophysics Data System (ADS)
Riseborough, Peter S.
2011-02-01
This article puts Korteweg and de Vries's manuscript (published in the Philosophical Magazine in 1895) in historical context. The article highlights the importance of the Korteweg-de Vries equation in the development of concepts used in nonlinear physics and also mentions some of their recent applications.
Mixed problems for the Korteweg-de Vries equation
Faminskii, A V
1999-06-30
Results are established concerning the non-local solubility and wellposedness in various function spaces of the mixed problem for the Korteweg-de Vries equation u{sub t}+u{sub xxx}+au{sub x}+uu{sub x}=f(t,x) in the half-strip (0,T)x(-{infinity},0). Some a priori estimates of the solutions are obtained using a special solution J(t,x) of the linearized KdV equation of boundary potential type. Properties of J are studied which differ essentially as x{yields}+{infinity} or x{yields}-{infinity}. Application of this boundary potential enables us in particular to prove the existence of generalized solutions with non-regular boundary values.
Interacting Korteweg-de Vries Equations and Attractive Soliton Interaction
NASA Astrophysics Data System (ADS)
Yoneyama, T.
1984-12-01
By a physically natural way, the Korteweg-de Vries (KdV) equation is extended to obtain textit{interacting} (Int) KdV equations. They can also be regarded as results of a decoupling of the original KdV equation. By introducing textit{new operators}, solutions of the Int KdV equations are obtained starting with the exact KdV N-soliton solution. The KdV N-soliton solution is decomposed into a textit{simple sum} of the solutions of the Int KdV equations, each of which is regarded as a soliton suffering much deformation when another soliton (other solitons) comes near in space. These single solitons as textit{classical waves} interact textit{attractively} and eventually become apart in space textit{without} losing their identities. The relation to the inverse scattering method is also discussed in detail. Further, ``partical'' Lax forms corresponding to the Int KdV equations are shown.
Nanopteron solution of the Korteweg-de Vries equation
NASA Astrophysics Data System (ADS)
Wang, Jianyong; Tang, Xiaoyan; Lou, Senyue; Gao, Xiaonan; Jia, Man
2014-10-01
The nanopteron, which is a permanent but weakly nonlocal soliton, has been an interesting topic in numerical studies for many decades. However, the analytical solution of such a special soliton is rarely considered. In this letter, we study the explicit nanopteron solution of the Korteweg-de Vries (KdV) equation. Starting from the soliton-cnoidal wave solution of the KdV equation, the nanopteron structure is shown to exist. It is found that for the suitable choice of the wave parameters, the soliton core of the soliton-cnoidal wave trends to be a classical soliton of the KdV equation and the surrounded cnoidal periodic wave appears as small amplitude sinusoidal variations on both sides of the main core. Some interesting features of the wave propagation are revealed. In addition to the elastic interaction, it is surprising that the phase shift of the cnoidal periodic wave after the interaction with the soliton core is always half its wavelength, and this conclusion is universal to soliton-cnoidal wave interactions.
On exact traveling-wave solutions for local fractional Korteweg-de Vries equation.
Yang, Xiao-Jun; Tenreiro Machado, J A; Baleanu, Dumitru; Cattani, Carlo
2016-08-01
This paper investigates the Korteweg-de Vries equation within the scope of the local fractional derivative formulation. The exact traveling wave solutions of non-differentiable type with the generalized functions defined on Cantor sets are analyzed. The results for the non-differentiable solutions when fractal dimension is 1 are also discussed. It is shown that the exact solutions for the local fractional Korteweg-de Vries equation characterize the fractal wave on shallow water surfaces.
Group analysis of general Burgers-Korteweg-de Vries equations
NASA Astrophysics Data System (ADS)
Opanasenko, Stanislav; Bihlo, Alexander; Popovych, Roman O.
2017-08-01
The complete group classification problem for the class of (1+1)-dimensional rth order general variable-coefficient Burgers-Korteweg-de Vries equations is solved for arbitrary values of r greater than or equal to two. We find the equivalence groupoids of this class and its various subclasses obtained by gauging equation coefficients with equivalence transformations. Showing that this class and certain gauged subclasses are normalized in the usual sense, we reduce the complete group classification problem for the entire class to that for the selected maximally gauged subclass, and it is the latter problem that is solved efficiently using the algebraic method of group classification. Similar studies are carried out for the two subclasses of equations with coefficients depending at most on the time or space variable, respectively. Applying an original technique, we classify Lie reductions of equations from the class under consideration with respect to its equivalence group. Studying alternative gauges for equation coefficients with equivalence transformations allows us not only to justify the choice of the most appropriate gauge for the group classification but also to construct for the first time classes of differential equations with nontrivial generalized equivalence group such that equivalence-transformation components corresponding to equation variables locally depend on nonconstant arbitrary elements of the class. For the subclass of equations with coefficients depending at most on the time variable, which is normalized in the extended generalized sense, we explicitly construct its extended generalized equivalence group in a rigorous way. The new notion of effective generalized equivalence group is introduced.
Negative-order Korteweg-de Vries equations.
Qiao, Zhijun; Fan, Engui
2012-07-01
In this paper, based on the regular Korteweg-de Vries (KdV) system, we study negative-order KdV (NKdV) equations, particularly their Hamiltonian structures, Lax pairs, conservation laws, and explicit multisoliton and multikink wave solutions thorough bilinear Bäcklund transformations. The NKdV equations studied in our paper are differential and actually derived from the first member in the negative-order KdV hierarchy. The NKdV equations are not only gauge equivalent to the Camassa-Holm equation through reciprocal transformations but also closely related to the Ermakov-Pinney systems and the Kupershmidt deformation. The bi-Hamiltonian structures and a Darboux transformation of the NKdV equations are constructed with the aid of trace identity and their Lax pairs, respectively. The single and double kink wave and bell soliton solutions are given in an explicit formula through the Darboux transformation. The one-kink wave solution is expressed in the form of tanh while the one-bell soliton is in the form of sech, and both forms are very standard. The collisions of two-kink wave and two-bell soliton solutions are analyzed in detail, and this singular interaction differs from the regular KdV equation. Multidimensional binary Bell polynomials are employed to find bilinear formulation and Bäcklund transformations, which produce N-soliton solutions. A direct and unifying scheme is proposed for explicitly building up quasiperiodic wave solutions of the NKdV equations. Furthermore, the relations between quasiperiodic wave solutions and soliton solutions are clearly described. Finally, we show the quasiperiodic wave solution convergent to the soliton solution under some limit conditions.
Soliton fractals in the Korteweg-de Vries equation.
Zamora-Sillero, Elias; Shapovalov, A V
2007-10-01
We have studied the process of creation of solitons and generation of fractal structures in the Korteweg-de Vries (KdV) equation when the relation between the nonlinearity and dispersion is abruptly changed. We observed that when this relation is changed nonadiabatically the solitary waves present in the system lose their stability and split up into ones that are stable for the set of parameters. When this process is successively repeated the trajectories of the solitary waves create a fractal treelike structure where each branch bifurcates into others. This structure is formed until the iteration where two solitary waves overlap just before the breakup. By means of a method based on the inverse scattering transformation, we have obtained analytical results that predict and control the number, amplitude, and velocity of the solitary waves that arise in the system after every change in the relation between the dispersion and the nonlinearity. This complete analytical information allows us to define a recursive L system which coincides with the treelike structure, governed by KdV, until the stage when the solitons start to overlap and is used to calculate the Hausdorff dimension and the multifractal properties of the set formed by the segments defined by each of the two "brothers" solitons before every breakup.
Similarity solutions of the generalized Korteweg-de Vries equation
NASA Astrophysics Data System (ADS)
Bona, J. L.; Weissler, F. B.
1999-09-01
Numerical simulations and group invariance considerations point to the existence of similarity solutions of the formformula hereof the generalized Korteweg-de Vries equationformula hereHere, x[low asterisk], t[low asterisk] and c are real parameters, x and t are real variables with t[not equal]t[low asterisk], p is a positive integer and interest is focussed on the case where p[gt-or-equal, slanted]4 for which solutions of the initial-value problem for (**) are not known to be always globally defined. It is shown that smooth solutions of (**) of the form appearing in (*) do indeed exist. Some detailed properties of the function [psi] appearing in (*) are also obtained.
Nonautonomous soliton solutions of the modified Korteweg-de Vries-sine-Gordon equation
NASA Astrophysics Data System (ADS)
Popov, S. P.
2016-11-01
Multisoliton solutions of the modified Korteweg-de Vries-sine-Gordon (mKdV-SG) equation with time-dependent coefficients are considered. Cases describing changes in the shape of soliton solutions (kinks and breathers) observed in gradual transitions between the mKdV, SG, and mKdV-SG equations are numerically studied.
Cylindrical nonlinear Schroedinger equation versus cylindrical Korteweg-de Vries equation
Fedele, Renato; De Nicola, Sergio; Grecu, Dan; Visinescu, Anca; Shukla, Padma K.
2008-10-15
A correspondence between the family of cylindrical nonlinear Schroedinger (cNLS) equations and the one of cylindrical Korteweg-de Vries (cKdV) equations is constructed. It associates non stationary solutions of the first family with the ones of the second family. This is done by using a correspondence, recently found, between the families of generalized NLS equation and generalized KdV equation, and their solutions in the form of travelling waves, respectively. In particular, non-stationary soliton-like solutions of the cNLS equation can be associated with non-stationary soliton-like solutions of cKdV equation.
1981-11-01
Cauchy problem for the Korteweg - deVries equation (KdV for short) q t(x,t) + q xx(x,t) - 6q(x,t)qx (x,t) = 0 q(x,0) =Q(x) - .is solved classically... KORTEWEG - deVRIES EQUATION FOR NON-SMOOTH INITIAL DATA VIA INVERSE SCATTERING Robert L. Sachs Technical Summary Report #2308 November 1981 ABSTRACT C The...1. SIGNIFICANCE AND EXPLANATION The Korteweg - deVries equation (KdV for short) arises as an approximation in many problems involving non-linear
An analytical method for finding exact solutions of modified Korteweg-de Vries equation
NASA Astrophysics Data System (ADS)
Islam, Md. Shafiqul; Khan, Kamruzzaman; Akbar, M. Ali
In this present work, we have studied new extension of the (G‧/G)-expansion method for finding the solitary wave solutions of the modified Korteweg-de Vries (mKdV) equation. It has been shown that the proposed method is effective and can be used for many other nonlinear evolution equations (NLEEs) in mathematical physics. The obtained results show that the method is very powerful and convenient mathematical tool for nonlinear evolution equations in science and engineering.
Formal initial value problem of the Korteweg-de Vries equation
NASA Astrophysics Data System (ADS)
Kim, Namhoon
2015-02-01
We study the initial value problem of the Korteweg-de Vries (KdV) equation on a space of generalized formal power series. We derive an explicit expression of the solution of the KdV equation with an arbitrary initial condition, using a recursively defined sequence of rational functions. From this result one can explain the formal analogues of the direct and inverse scattering transforms, relating the given initial condition to the solution of the formal Gelfand-Levitan-Marchenko equation.
The zero dispersion limit for the Korteweg-deVries KdV equation
Lax, Peter D.; Levermore, C. David
1979-01-01
We use the inverse scattering method to determine the weak limit of solutions of the Korteweg-deVries equation as dispersion tends to zero. The limit, valid for all time, is characterized in terms of a quadratic programming problem which can be solved with the aid of function theoretic methods. For large t, the solutions satisfy Whitham's averaged equations at some times and the equations found by Flaschka et al. at other times. PMID:16592690
Darboux transformation for a generalized Hirota-Satsuma coupled Korteweg-de Vries equation
Geng Xianguo; Ren Hongfeng; He Guoliang
2009-05-15
A Darboux transformation for the generalized Hirota-Satsuma coupled Korteweg-de Vries (KdV) equation is derived with the aid of the gauge transformation between the corresponding 4x4 matrix spectral problems with three potentials, by which some explicit solutions of the generalized Hirota-Satsuma coupled KdV equation are constructed. As a reduction, a Darboux transformation of the complex coupled KdV equation and its explicit solutions are obtained.
The zero dispersion limit for the Korteweg-deVries KdV equation.
Lax, P D; Levermore, C D
1979-08-01
We use the inverse scattering method to determine the weak limit of solutions of the Korteweg-deVries equation as dispersion tends to zero. The limit, valid for all time, is characterized in terms of a quadratic programming problem which can be solved with the aid of function theoretic methods. For large t, the solutions satisfy Whitham's averaged equations at some times and the equations found by Flaschka et al. at other times.
Darboux transformation for a generalized Hirota-Satsuma coupled Korteweg-de Vries equation.
Geng, Xianguo; Ren, Hongfeng; He, Guoliang
2009-05-01
A Darboux transformation for the generalized Hirota-Satsuma coupled Korteweg-de Vries (KdV) equation is derived with the aid of the gauge transformation between the corresponding 4x4 matrix spectral problems with three potentials, by which some explicit solutions of the generalized Hirota-Satsuma coupled KdV equation are constructed. As a reduction, a Darboux transformation of the complex coupled KdV equation and its explicit solutions are obtained.
The Sylvester equation and the elliptic Korteweg-de Vries system
NASA Astrophysics Data System (ADS)
Sun, Ying-ying; Zhang, Da-jun; Nijhoff, Frank W.
2017-03-01
The elliptic potential Korteweg-de Vries lattice system is a multi-component extension of the lattice potential Korteweg-de Vries equation, whose soliton solutions are associated with an elliptic Cauchy kernel (i.e., a Cauchy kernel on the torus). In this paper we generalize the class of solutions by allowing the spectral parameter to be a full matrix obeying a matrix version of the equation of the elliptic curve, and for the Cauchy matrix to be a solution of a Sylvester type matrix equation subject to this matrix elliptic curve equation. The construction involves solving the matrix elliptic curve equation by using Toeplitz matrix techniques, and analysing the solution of the Sylvester equation in terms of Jordan normal forms. Furthermore, we consider the continuum limit system associated with the elliptic potential Korteweg-de Vries system, and analyse the dynamics of the soliton solutions, which reveals some new features of the elliptic system in comparison to the non-elliptic case.
Asymptotic stability of a nonlinear Korteweg-de Vries equation with critical lengths
NASA Astrophysics Data System (ADS)
Chu, Jixun; Coron, Jean-Michel; Shang, Peipei
2015-10-01
We study an initial-boundary-value problem of a nonlinear Korteweg-de Vries equation posed on the finite interval (0, 2 kπ) where k is a positive integer. The whole system has Dirichlet boundary condition at the left end-point, and both of Dirichlet and Neumann homogeneous boundary conditions at the right end-point. It is known that the origin is not asymptotically stable for the linearized system around the origin. We prove that the origin is (locally) asymptotically stable for the nonlinear system if the integer k is such that the kernel of the linear Korteweg-de Vries stationary equation is of dimension 1. This is for example the case if k = 1.
Breathers and localized solutions of complex modified Korteweg-de Vries equation
NASA Astrophysics Data System (ADS)
Liu, Yue-Feng; Guo, Rui; Li, Hua
2015-07-01
Under investigation is the complex modified Korteweg-de Vries (KdV) equation, which has many physical significance in fluid mechanics, plasma physics and so on. Via the Darboux transformation (DT) method, some breather and localized solutions are presented on two backgrounds: the continuous wave background u1 = kexp[i(Ax + Bt)] and the constant background u2 = a + ib. Some figures are plotted to illustrate the dynamical features of those solutions.
Error analysis for spectral approximation of the Korteweg-De Vries equation
NASA Technical Reports Server (NTRS)
Maday, Y.; recent years.
1987-01-01
The conservation and convergence properties of spectral Fourier methods for the numerical approximation of the Korteweg-de Vries equation are analyzed. It is proved that the (aliased) collocation pseudospectral method enjoys the same convergence properties as the spectral Galerkin method, which is less effective from the computational point of view. This result provides a precise mathematical answer to a question raised by several authors in recent years.
Nonlinear dynamics of a soliton gas: Modified Korteweg-de Vries equation framework
NASA Astrophysics Data System (ADS)
Shurgalina, E. G.; Pelinovsky, E. N.
2016-05-01
Dynamics of random multi-soliton fields within the framework of the modified Korteweg-de Vries equation is considered. Statistical characteristics of a soliton gas (distribution functions and moments) are calculated. It is demonstrated that the results sufficiently depend on the soliton gas properties, i.e., whether it is unipolar or bipolar. It is shown that the properties of a unipolar gas are qualitatively similar to the properties of a KdV gas [Dutykh and Pelinovsky (2014) [1
Zhou Ruguang
2007-01-15
A procedure of nonlinearization of spectral problem that allows to impose reality conditions or restriction conditions on potentials is presented. As applications, integrable decompositions of the nonlinear Schroedinger equation and the real-valued modified Korteweg-de Vries equation are obtained.
NASA Astrophysics Data System (ADS)
Ji, Jia-Liang; Zhu, Zuo-Nong
2017-01-01
Very recently, Ablowitz and Musslimani introduced a new integrable nonlocal nonlinear Schrödinger equation. In this paper, we investigate an integrable nonlocal modified Korteweg-de Vries equation (mKdV) which can be derived from the well-known AKNS system. We construct the Darboux transformation for the nonlocal mKdV equation. Using the Darboux transformation, we obtain its different kinds of exact solutions including soliton, kink, antikink, complexiton, rogue-wave solution, and nonlocalized solution with singularities. It is shown that these solutions possess new properties which are different from the ones for mKdV equation.
Periodic and rational solutions of modified Korteweg-de Vries equation
NASA Astrophysics Data System (ADS)
Chowdury, Amdad; Ankiewicz, Adrian; Akhmediev, Nail
2016-05-01
We present closed form periodic solutions of the integrable modified Korteweg-de Vries equation (mKdV). By using a Darboux transformation, we derive first-and second-order doubly-periodic lattice-like solutions. We explicitly derive first-and second-order rational solutions as limiting cases of periodic solutions. We have also found the degenerate solution which corresponds to the equal eigenvalue case. Among the second-order solutions, we single out the doubly-localized high peak solution on a constant background with an infinitely extended trough. This solution plays the role of a rogue wave of the mKdV equation.
Residual Symmetries and Interaction Solutions for the Classical Korteweg-de Vries Equation
NASA Astrophysics Data System (ADS)
Fei, Jin-Xi; Cao, Wei-Ping; Ma, Zheng-Yi
2017-03-01
The non-local residual symmetry for the classical Korteweg-de Vries equation is derived by the truncated Painlevé analysis. This symmetry is first localised to the Lie point symmetry by introducing the auxiliary dependent variables. By using Lie's first theorem, we then obtain the finite transformation for the localised residual symmetry. Based on the consistent tanh expansion method, some exact interaction solutions among different non-linear excitations are explicitly presented finally. Some special interaction solutions are investigated both in analytical and graphical ways at the same time.
Exact solutions of unsteady Korteweg-de Vries and time regularized long wave equations.
Islam, S M Rayhanul; Khan, Kamruzzaman; Akbar, M Ali
2015-01-01
In this paper, we implement the exp(-Φ(ξ))-expansion method to construct the exact traveling wave solutions for nonlinear evolution equations (NLEEs). Here we consider two model equations, namely the Korteweg-de Vries (KdV) equation and the time regularized long wave (TRLW) equation. These equations play significant role in nonlinear sciences. We obtained four types of explicit function solutions, namely hyperbolic, trigonometric, exponential and rational function solutions of the variables in the considered equations. It has shown that the applied method is quite efficient and is practically well suited for the aforementioned problems and so for the other NLEEs those arise in mathematical physics and engineering fields. PACS numbers: 02.30.Jr, 02.70.Wz, 05.45.Yv, 94.05.Fq.
Dynamics of soliton fields in the framework of modified Korteweg - de Vries equation
NASA Astrophysics Data System (ADS)
Pelinovsky, Efim; Shurgalina, Ekaterina
2014-05-01
The dynamics of soliton field in the framework of modified Korteweg-de Vries (mKdV) equation is studied. Two-soliton interactions play a definitive role in the formation of the structure of soliton field. Three types of soliton interaction are considered: exchange and overtaking for solitons of the same polarity, and absorb-emit for solitons of different polarity. Features of soliton interaction are studied in details. Since the interaction of solitons is an elementary act of soliton turbulence, the moments of the wave field up to fourth are studied, which are usually applied in the turbulence theory. It is shown that in the case of interaction of solitons of the same polarity the third and fourth moments of the wave field, which determine the coefficients of skewness and kurtosis in the turbulence theory, are reduced, while in the case of interaction of solitons of different polarity these moments are increased. Numerical study of the statistical characteristics of multi-soliton fields which are generated from the initially isolated solitons with random phases and amplitudes is made. The effect of the nonlinear interaction between solitons and dispersive trains is analysed. It is confirmed that first two moments being the invariants of the modified Korteweg - de Vries equation remain to be constant. The skewness and kurtosis vary in time in each realization but tends to the constants in the average.
NASA Astrophysics Data System (ADS)
Johnpillai, Andrew G.; Kara, Abdul H.; Biswas, Anjan
2013-09-01
We study the scalar complex modified Korteweg-de Vries (cmKdV) equation by analyzing a system of partial differential equations (PDEs) from the Lie symmetry point of view. These systems of PDEs are obtained by decomposing the underlying cmKdV equation into real and imaginary components. We derive the Lie point symmetry generators of the system of PDEs and classify them to get the optimal system of one-dimensional subalgebras of the Lie symmetry algebra of the system of PDEs. These subalgebras are then used to construct a number of symmetry reductions and exact group invariant solutions to the system of PDEs. Finally, using the Lie symmetry approach, a couple of new conservation laws are constructed. Subsequently, respective conserved quantities from their respective conserved densities are computed.
Decay of Solutions to Damped Korteweg-de Vries Type Equation
Cavalcanti, Marcelo M. Domingos Cavalcanti, Valeria N.; Faminskii, Andrei; Natali, Fabio
2012-04-15
In the present paper we establish results concerning the decay of the energy related to the damped Korteweg-de Vries equation posed on infinite domains. We prove the exponential decay rates of the energy when a initial value problem and a localized dissipative mechanism are in place. If this mechanism is effective in the whole line, we get a similar result in H{sup k}-level, k Element-Of Double-Struck-Capital-N . In addition, the decay of the energy regarding a initial boundary value problem posed on the right half-line, is obtained considering convenient a smallness condition on the initial data but a more general dissipative effect.
Interactions of breathers and solitons of the extended Korteweg de Vries equation
NASA Astrophysics Data System (ADS)
Shek, C. M.; Grimshaw, R. H. J.; Ding, E.
2005-11-01
A popular model for the evolution of weakly nonlinear, weakly dispersive waves in the ocean is the extended Korteweg -- de Vries equation (eKdV), which incorporates both quadratic and cubic nonlinearities. The case of positive cubic nonlinearity allows for both solitons of elevation and depression, as well as breathers (pulsating modes). Multi-soliton solutions are computed analytically, and will yield expressions for breather-soliton interactions. Both the soliton and breather will retain their identities after interactions, but suffer phase shifts. However, the details of the interaction process will depend on the polarity of the interacting soliton, and have been investigated by a computer algebra software. This highly time dependent motion during the interaction process is important in nonlinear science and physical oceanography. As the dynamics of the current and an evolving internal oceanic tide can be modeled by eKdV, this knowledge is relevant to the temporal and spatial variability observed in the oceanic internal soliton fields.
Korteweg-de Vries Burgers equation for magnetosonic wave in plasma
Hussain, S.; Mahmood, S.
2011-05-15
Korteweg-de Vries Burgers (KdVB) equation for magnetosonic wave propagating in the perpendicular direction of the magnetic field is derived for homogeneous electron-ion magneto-plasmas. The dissipation in the system is taken into account through the kinematic viscosity of the ions. The effects of kinematic viscosity of ions, plasma density, and magnetic field strength on the formation of magnetosonic shocks are investigated. It is found that the shock strength is enhanced with the increase in the plasma density of the system. However, the increase in magnetic field strength decreases the amplitude of magnetosonic shock wave. The critical value of the dissipative coefficient to form oscillatory profile and monotonic shock is also discussed. The numerical results have also been plotted by taking the parameters from laboratory plasma experiments.
Compacton solutions in a class of generalized fifth-order Korteweg-de Vries equations.
Cooper, F; Hyman, J M; Khare, A
2001-08-01
Solitons play a fundamental role in the evolution of general initial data for quasilinear dispersive partial differential equations, such as the Korteweg-de Vries (KdV), nonlinear Schrödinger, and the Kadomtsev-Petviashvili equations. These integrable equations have linear dispersion and the solitons have infinite support. We have derived and investigate a new KdV-like Hamiltonian partial differential equation from a four-parameter Lagrangian where the nonlinear dispersion gives rise to solitons with compact support (compactons). The new equation does not seem to be integrable and only mass, momentum, and energy seem to be conserved; yet, the solitons display almost the same modal decompositions and structural stability observed in integrable partial differential equations. The compactons formed from arbitrary initial data, are nonlinearly self-stabilizing, and maintain their coherence after multiple collisions. The robustness of these compactons and the inapplicability of the inverse scattering tools, that worked so well for the KdV equation, make it clear that there is a fundamental mechanism underlying the processes beyond integrability. We have found explicit formulas for multiple classes of compact traveling wave solutions. When there are more than one compacton solution for a particular set of parameters, the wider compacton is the minimum of a reduced Hamiltonian and is the only one that is stable.
Compacton solutions in a class of generalized fifth-order Korteweg--de Vries equations
Cooper, Fred; Hyman, James M.; Khare, Avinash
2001-08-01
Solitons play a fundamental role in the evolution of general initial data for quasilinear dispersive partial differential equations, such as the Korteweg--de Vries (KdV), nonlinear Schroedinger, and the Kadomtsev-Petviashvili equations. These integrable equations have linear dispersion and the solitons have infinite support. We have derived and investigate a new KdV-like Hamiltonian partial differential equation from a four-parameter Lagrangian where the nonlinear dispersion gives rise to solitons with compact support (compactons). The new equation does not seem to be integrable and only mass, momentum, and energy seem to be conserved; yet, the solitons display almost the same modal decompositions and structural stability observed in integrable partial differential equations. The compactons formed from arbitrary initial data, are nonlinearly self-stabilizing, and maintain their coherence after multiple collisions. The robustness of these compactons and the inapplicability of the inverse scattering tools, that worked so well for the KdV equation, make it clear that there is a fundamental mechanism underlying the processes beyond integrability. We have found explicit formulas for multiple classes of compact traveling wave solutions. When there are more than one compacton solution for a particular set of parameters, the wider compacton is the minimum of a reduced Hamiltonian and is the only one that is stable.
Soliton evolution and radiation loss for the Korteweg--de Vries equation
Kath, W.L.; Smyth, N.F. Department of Mathematics and Statistics, University of Edinburgh, The King's Buildings, Mayfield Road, Edinburgh EH93JZ, Scotland )
1995-01-01
The time-dependent behavior of solutions of the Korteweg--de Vries (KdV) equation for nonsoliton initial conditions is considered. While the exact solution of the KdV equation can in principle be obtained using the inverse scattering transform, in practice it can be extremely difficult to obtain information about a solution's transient evolution by this method. As an alternative, we present here an approximate method for investigating this transient evolution which is based upon the conservation laws associated with the KdV equation. Initial conditions which form one or two solitons are considered, and the resulting approximate evolution is found to be in good agreement with the numerical solution of the KdV equation. Justification for the approximations employed is also given by way of the linearized inverse scattering solution of the KdV equation. In addition, the final soliton state determined from the approximate equations agrees very well with the final state determined from the exact inverse scattering transform solution.
Kahan, W.; Li, Ren-Chang
1997-07-01
An unconventional numerical method for solving a restrictive and yet often-encountered class of ordinary differential equations is proposed. The method has a crucial, what we call reflexive, property and requires solving one linear system per time-step, but is second-order accurate. A systematical and easily implementable scheme is proposed to enhance the computational efficiency of such methods whenever needed. Applications are reported on how the idea can be applied to solve the Korteweg-de Vries Equation discretized in space.
NASA Astrophysics Data System (ADS)
Liu, Hailiang; Yi, Nianyu
2016-09-01
The invariant preserving property is one of the guiding principles for numerical algorithms in solving wave equations, in order to minimize phase and amplitude errors after long time simulation. In this paper, we design, analyze and numerically validate a Hamiltonian preserving discontinuous Galerkin method for solving the Korteweg-de Vries (KdV) equation. For the generalized KdV equation, the semi-discrete formulation is shown to preserve both the first and the third conserved integrals, and approximately preserve the second conserved integral; for the linearized KdV equation, all the first three conserved integrals are preserved, and optimal error estimates are obtained for polynomials of even degree. The preservation properties are also maintained by the fully discrete DG scheme. Our numerical experiments demonstrate both high accuracy of convergence and preservation of all three conserved integrals for the generalized KdV equation. We also show that the shape of the solution, after long time simulation, is well preserved due to the Hamiltonian preserving property.
A simple and robust boundary treatment for the forced Korteweg-de Vries equation
NASA Astrophysics Data System (ADS)
Lee, Hyun Geun; Kim, Junseok
2014-07-01
In this paper, we propose a simple and robust numerical method for the forced Korteweg-de Vries (fKdV) equation which models free surface waves of an incompressible and inviscid fluid flow over a bump. The fKdV equation is defined in an infinite domain. However, to solve the equation numerically we must truncate the infinite domain to a bounded domain by introducing an artificial boundary and imposing boundary conditions there. Due to unsuitable artificial boundary conditions, most wave propagation problems have numerical difficulties (e.g., the truncated computational domain must be large enough or the numerical simulation must be terminated before the wave approaches the artificial boundary for the quality of the numerical solution). To solve this boundary problem, we develop an absorbing non-reflecting boundary treatment which uses outward wave velocity. The basic idea of the proposing algorithm is that we first calculate an outward wave velocity from the solutions at the previous and present time steps and then we obtain a solution at the next time step on the artificial boundary by moving the solution at the present time step with the velocity. And then we update solutions at the next time step inside the domain using the calculated solution on the artificial boundary. Numerical experiments with various initial conditions for the KdV and fKdV equations are presented to illustrate the accuracy and efficiency of our method.
Shallow-water soliton dynamics beyond the Korteweg-de Vries equation.
Karczewska, Anna; Rozmej, Piotr; Infeld, Eryk
2014-07-01
An alternative way for the derivation of the new Korteweg-de Vries (KdV)-type equation is presented. The equation contains terms depending on the bottom topography (there are six new terms in all, three of which are caused by the unevenness of the bottom). It is obtained in the second-order perturbative approach in the weakly nonlinear, dispersive, and long wavelength limit. Only treating all these terms in the second-order perturbation theory made the derivation of this KdV-type equation possible. The motion of a wave, which starts as a KdV soliton, is studied according to the new equation in several cases by numerical simulations. The quantitative changes of a soliton's velocity and amplitude appear to be directly related to bottom variations. Changes of the soliton's velocity appear to be almost linearly anticorrelated with changes of water depth, whereas correlation of variation of soliton's amplitude with changes of water depth looks less linear. When the bottom is flat, the new terms narrow down the family of exact solutions, but at least one single soliton survives. This is also checked by numerics.
Shallow-water soliton dynamics beyond the Korteweg-de Vries equation
NASA Astrophysics Data System (ADS)
Karczewska, Anna; Rozmej, Piotr; Infeld, Eryk
2014-07-01
An alternative way for the derivation of the new Korteweg-de Vries (KdV)-type equation is presented. The equation contains terms depending on the bottom topography (there are six new terms in all, three of which are caused by the unevenness of the bottom). It is obtained in the second-order perturbative approach in the weakly nonlinear, dispersive, and long wavelength limit. Only treating all these terms in the second-order perturbation theory made the derivation of this KdV-type equation possible. The motion of a wave, which starts as a KdV soliton, is studied according to the new equation in several cases by numerical simulations. The quantitative changes of a soliton's velocity and amplitude appear to be directly related to bottom variations. Changes of the soliton's velocity appear to be almost linearly anticorrelated with changes of water depth, whereas correlation of variation of soliton's amplitude with changes of water depth looks less linear. When the bottom is flat, the new terms narrow down the family of exact solutions, but at least one single soliton survives. This is also checked by numerics.
Ratliff, Daniel J; Bridges, Thomas J
2016-12-01
Multiphase wavetrains are multiperiodic travelling waves with a set of distinct wavenumbers and distinct frequencies. In conservative systems, such families are associated with the conservation of wave action or other conservation law. At generic points (where the Jacobian of the wave action flux is non-degenerate), modulation of the wavetrain leads to the dispersionless multiphase conservation of wave action. The main result of this paper is that modulation of the multiphase wavetrain, when the Jacobian of the wave action flux vector is singular, morphs the vector-valued conservation law into the scalar Korteweg-de Vries (KdV) equation. The coefficients in the emergent KdV equation have a geometrical interpretation in terms of projection of the vector components of the conservation law. The theory herein is restricted to two phases to simplify presentation, with extensions to any finite dimension discussed in the concluding remarks. Two applications of the theory are presented: a coupled nonlinear Schrödinger equation and two-layer shallow-water hydrodynamics with a free surface. Both have two-phase solutions where criticality and the properties of the emergent KdV equation can be determined analytically.
Quartic B-spline collocation method applied to Korteweg de Vries equation
NASA Astrophysics Data System (ADS)
Zin, Shazalina Mat; Majid, Ahmad Abd; Ismail, Ahmad Izani Md
2014-07-01
The Korteweg de Vries (KdV) equation is known as a mathematical model of shallow water waves. The general form of this equation is ut+ɛuux+μuxxx = 0 where u(x,t) describes the elongation of the wave at displacement x and time t. In this work, one-soliton solution for KdV equation has been obtained numerically using quartic B-spline collocation method for displacement x and using finite difference approach for time t. Two problems have been identified to be solved. Approximate solutions and errors for these two test problems were obtained for different values of t. In order to look into accuracy of the method, L2-norm and L∞-norm have been calculated. Mass, energy and momentum of KdV equation have also been calculated. The results obtained show the present method can approximate the solution very well, but as time increases, L2-norm and L∞-norm are also increase.
Derivation of electrostatic Korteweg-deVries equation in fully relativistic two-fluid plasmas
Lee, Nam C.
2008-08-15
A second order Korteweg-deVries (KdV) equation that describes the evolution of nonlinear electrostatic waves in fully relativistic two-fluid plasmas is derived without any assumptions restricting the magnitudes of the flow velocity and the temperatures of each species. In the derivation, the positive and negative species of plasmas are treated with equal footings, not making any species specific assumptions. Thus, the resulting equation, which is expressed in transparent form symmetric in particle species, can be applied to any two-fluid plasmas having arbitrarily large flow velocity and ultrarelativistically high temperatures. The phase velocity of the nonlinear electrostatic waves found in this paper is shown to be related to the flow velocity and the acoustic wave velocity through the Lorentz addition law of velocities, revealing the relativistic nature of the formulation in the present study. The derived KdV equation is applied to some limiting cases, and it is shown that it can be reduced to existing results in nonrelativistic plasmas, while there are some discrepancies from the results in the weak relativistic approximations.
Few-cycle optical rogue waves: complex modified Korteweg-de Vries equation.
He, Jingsong; Wang, Lihong; Li, Linjing; Porsezian, K; Erdélyi, R
2014-06-01
In this paper, we consider the complex modified Korteweg-de Vries (mKdV) equation as a model of few-cycle optical pulses. Using the Lax pair, we construct a generalized Darboux transformation and systematically generate the first-, second-, and third-order rogue wave solutions and analyze the nature of evolution of higher-order rogue waves in detail. Based on detailed numerical and analytical investigations, we classify the higher-order rogue waves with respect to their intrinsic structure, namely, fundamental pattern, triangular pattern, and ring pattern. We also present several new patterns of the rogue wave according to the standard and nonstandard decomposition. The results of this paper explain the generalization of higher-order rogue waves in terms of rational solutions. We apply the contour line method to obtain the analytical formulas of the length and width of the first-order rogue wave of the complex mKdV and the nonlinear Schrödinger equations. In nonlinear optics, the higher-order rogue wave solutions found here will be very useful to generate high-power few-cycle optical pulses which will be applicable in the area of ultrashort pulse technology.
NASA Astrophysics Data System (ADS)
Sharma, Dinkar; Singh, Prince; Chauhan, Shubha
2017-06-01
In this paper, a combined form of the Laplace transform method with the homotopy perturbation method is applied to solve nonlinear fifth order Korteweg de Vries (KdV) equations. The method is known as homotopy perturbation transform method (HPTM). The nonlinear terms can be easily handled by the use of He's polynomials. Two test examples are considered to illustrate the present scheme. Further the results are compared with Homotopy perturbation method (HPM).
Rare-event Simulation for Stochastic Korteweg-de Vries Equation
Xu, Gongjun; Lin, Guang; Liu, Jingchen
2014-01-01
An asymptotic analysis of the tail probabilities for the dynamics of a soliton wave $U(x,t)$ under a stochastic time-dependent force is developed. The dynamics of the soliton wave $U(x,t)$ is described by the Korteweg-de Vries Equation with homogeneous Dirichlet boundary conditions under a stochastic time-dependent force, which is modeled as a time-dependent Gaussian noise with amplitude $\\epsilon$. The tail probability we considered is $w(b) :=P(\\sup_{t\\in [0,T]} U(x,t) > b ),$ as $b\\rightarrow \\infty,$ for some constant $T>0$ and a fixed $x$, which can be interpreted as tail probability of the amplitude of water wave on shallow surface of a fluid or long internal wave in a density-stratified ocean. Our goal is to characterize the asymptotic behaviors of $w(b)$ and to evaluate the tail probability of the event that the soliton wave exceeds a certain threshold value under a random force term. Such rare-event calculation of $w(b)$ is very useful for fast estimation of the risk of the potential damage that could caused by the water wave in a density-stratified ocean modeled by the stochastic KdV equation. In this work, the asymptotic approximation of the probability that the soliton wave exceeds a high-level $b$ is derived. In addition, we develop a provably efficient rare-event simulation algorithm to compute $w(b)$. The efficiency of the algorithm only requires mild conditions and therefore it is applicable to a general class of Gaussian processes and many diverse applications.
Karczewska, Anna; Rozmej, Piotr; Infeld, Eryk
2015-11-01
It is well known that the Korteweg-de Vries (KdV) equation has an infinite set of conserved quantities. The first three are often considered to represent mass, momentum, and energy. Here we try to answer the question of how this comes about and also how these KdV quantities relate to those of the Euler shallow-water equations. Here Luke's Lagrangian is helpful. We also consider higher-order extensions of KdV. Though in general not integrable, in some sense they are almost so within the accuracy of the expansion.
NASA Astrophysics Data System (ADS)
Basu, Ashis; Ray, Dipankar
1990-05-01
In a recent paper, Moreira [Res. Rep. IF/UFRJ/83/25, Universidade Federal do Rio de Janeiro Inst. de Fisica, Cidade Univ., Ilha do Fundao, Rio de Janeiro, Brazil] obtained a nonlinear second-order differential equation that leads to the first integral of a modified Emden equation. He also obtained two particular solutions of his equation. This paper completely integrates Moreira's equation and uses it to get a class of solutions of a coupled Korteweg-deVries (KdV) equation, recently studied by Guha Ray, Bagchi, and Sinha [J. Math. Phys. 27, 2558 (1986)].
NASA Astrophysics Data System (ADS)
Restuccia, Alvaro; Sotomayor, Adrián
2016-08-01
We present a local Bäcklund Wahlquist-Estabrook (WE) transformation for a supersymmetric Korteweg-de Vries (KdV) equation. As in the scalar case, such type of transformation generates infinite hierarchies of solutions and also implicitly gives the associated (local) conserved quantities. A nice property is that every of such hierarchies admits a nonlinear superposition principle, starting for an initial solution, including as a particular case the multisolitonic solutions of the system. We discuss the symmetries of the system and we present in an explicit way its local conserved quantities with the help of the associated Gardner transformation.
Korteweg de Vries Burgers equation in multi-ion and pair-ion plasmas with Lorentzian electrons
Hussain, S.; Akhtar, N.
2013-01-15
Korteweg de Vries Burgers equation for multi-ion and pair-ion plasmas has been derived using reductive perturbation technique. The kinematic viscosities of both positive and negative ions are taken into account. Generalized Lorentzian distribution is assumed for the electron component, accounting for deviation from Maxwellian equilibrium, parametrized via a real parameter {kappa}. The modification in the strength of shock structure is presented. A comprehensive comparison between the profiles of shock wave structure in multi-ion and pair-ion plasmas, (for the Maxwellian electrons to Lorentzian electrons), is discussed.
Supersymmetric quantum mechanics and the Korteweg--de Vries hierarchy
Grant, A.K.; Rosner, J.L. )
1994-05-01
The connection between supersymmetric quantum mechanics and the Korteweg--de Vries (KdV) equation is discussed, with particular emphasis on the KdV conservation laws. It is shown that supersymmetric quantum mechanics aids in the derivation of the conservation laws, and gives some insight into the Miura transformation that converts the KdV equation into the modified KdV equation. The construction of the [tau] function by means of supersymmetric quantum mechanics is discussed.
On a hierarchy of nonlinearly dispersive generalized Korteweg - de Vries evolution equations
Christov, Ivan C.
2015-08-20
We propose a hierarchy of nonlinearly dispersive generalized Korteweg–de Vries (KdV) evolution equations based on a modification of the Lagrangian density whose induced action functional the KdV equation extremizes. Two recent nonlinear evolution equations describing wave propagation in certain generalized continua with an inherent material length scale are members of the proposed hierarchy. Like KdV, the equations from the proposed hierarchy possess Hamiltonian structure. Unlike KdV, the solutions to these equations can be compact (i.e., they vanish outside of some open interval) and, in addition, peaked. Implicit solutions for these peaked, compact traveling waves (“peakompactons”) are presented.
Hussin, V.; Kiselev, A. V.; Krutov, A. O.; Wolf, T.
2010-08-15
We consider the problem of constructing Gardner's deformations for the N=2 supersymmetric a=4-Korteweg-de Vries (SKdV) equation; such deformations yield recurrence relations between the super-Hamiltonians of the hierarchy. We prove the nonexistence of supersymmetry-invariant deformations that retract to Gardner's formulas for the Korteweg-de Vries (KdV) with equation under the component reduction. At the same time, we propose a two-step scheme for the recursive production of the integrals of motion for the N=2, a=4-SKdV. First, we find a new Gardner's deformation of the Kaup-Boussinesq equation, which is contained in the bosonic limit of the superhierarchy. This yields the recurrence relation between the Hamiltonians of the limit, whence we determine the bosonic super-Hamiltonians of the full N=2, a=4-SKdV hierarchy. Our method is applicable toward the solution of Gardner's deformation problems for other supersymmetric KdV-type systems.
NASA Astrophysics Data System (ADS)
Klein, C.; Peter, R.
2015-06-01
We present a detailed numerical study of solutions to general Korteweg-de Vries equations with critical and supercritical nonlinearity, both in the context of dispersive shocks and blow-up. We study the stability of solitons and show that they are unstable against being radiated away and blow-up. In the L2 critical case, the blow-up mechanism by Martel, Merle and Raphaël can be numerically identified. In the limit of small dispersion, it is shown that a dispersive shock always appears before an eventual blow-up. In the latter case, always the first soliton to appear will blow up. It is shown that the same type of blow-up as for the perturbations of the soliton can be observed which indicates that the theory by Martel, Merle and Raphaël is also applicable to initial data with a mass much larger than the soliton mass. We study the scaling of the blow-up time t∗ in dependence of the small dispersion parameter ɛ and find an exponential dependence t∗(ɛ) and that there is a minimal blow-up time t0∗ greater than the critical time of the corresponding Hopf solution for ɛ → 0. To study the cases with blow-up in detail, we apply the first dynamic rescaling for generalized Korteweg-de Vries equations. This allows to identify the type of the singularity.
Spontaneous soliton generation in the higher order Korteweg-de Vries equations on the half-line.
Burde, G I
2012-03-01
Some new effects in the soliton dynamics governed by higher order Korteweg-de Vries (KdV) equations are discussed based on the exact explicit solutions of the equations on the positive half-line. The solutions describe the process of generation of a soliton that occurs without boundary forcing and on the steady state background: the boundary conditions remain constant and the initial distribution is a steady state solution of the problem. The time moment when the soliton generation starts is not determined by the parameters present in the problem formulation, the additional parameters imbedded into the solution are needed to determine that moment. The equations found capable of describing those effects are the integrable Sawada-Kotera equation and the KdV-Kaup-Kupershmidt (KdV-KK) equation which, albeit not proven to be integrable, possesses multi-soliton solutions.
NASA Astrophysics Data System (ADS)
Wang, Huimin
2017-01-01
In this paper, a new lattice Boltzmann model for the Korteweg-de Vries (KdV) equation is proposed. By using the Chapman-Enskog expansion and the multi-scale time expansion, a series of partial differential equations in different time scales and several higher- order moments of equilibrium distribution functions are obtained. In order to make the scheme obey the three conservation laws of the KdV equation, two equilibrium distribution functions are used and a correlation between the first conservation law and the second conservation law is constructed. In numerical examples, the numerical results of the KdV equation obtained by this scheme are compared with those results obtained by the previous lattice Boltzmann model. Numerical experiments demonstrate this scheme can be used to reduce the truncation error of the lattice Boltzmann scheme and preserve the three conservation laws.
Explicit solutions from eigenfunction symmetry of the Korteweg-de Vries equation.
Hu, Xiao-Rui; Lou, Sen-Yue; Chen, Yong
2012-05-01
In nonlinear science, it is very difficult to find exact interaction solutions among solitons and other kinds of complicated waves such as cnoidal waves and Painlevé waves. Actually, even if for the most well-known prototypical models such as the Kortewet-de Vries (KdV) equation and the Kadomtsev-Petviashvili (KP) equation, this kind of problem has not yet been solved. In this paper, the explicit analytic interaction solutions between solitary waves and cnoidal waves are obtained through the localization procedure of nonlocal symmetries which are related to Darboux transformation for the well-known KdV equation. The same approach also yields some other types of interaction solutions among different types of solutions such as solitary waves, rational solutions, Bessel function solutions, and/or general Painlevé II solutions.
Guo Shimin; Wang Hongli; Mei Liquan
2012-06-15
By combining the effects of bounded cylindrical geometry, azimuthal and axial perturbations, the nonlinear dust acoustic waves (DAWs) in an unmagnetized plasma consisting of negatively charged dust grains, nonextensive ions, and nonextensive electrons are studied in this paper. Using the reductive perturbation method, a (3 + 1)-dimensional variable-coefficient cylindrical Korteweg-de Vries (KdV) equation describing the nonlinear propagation of DAWs is derived. Via the homogeneous balance principle, improved F-expansion technique and symbolic computation, the exact traveling and solitary wave solutions of the KdV equation are presented in terms of Jacobi elliptic functions. Moreover, the effects of the plasma parameters on the solitary wave structures are discussed in detail. The obtained results could help in providing a good fit between theoretical analysis and real applications in space physics and future laboratory plasma experiments where long-range interactions are present.
NASA Astrophysics Data System (ADS)
Qin, Chun-Yan; Tian, Shou-Fu; Wang, Xiu-Bin; Zhang, Tian-Tian
2016-07-01
Under investigation in this paper is a fifth-order Korteweg-de Vries type (fKdV-type) equation with time-dependent coefficients, which can be used to describe many nonlinear phenomena in fluid mechanics, ocean dynamics and plasma physics. The binary Bell polynomials are employed to find its Hirota’s bilinear formalism with an extra auxiliary variable, based on which its N-soliton solutions can be also directly derived. Furthermore, by considering multi-dimensional Riemann theta function, a lucid and straightforward generalization of the Hirota-Riemann method is presented to explicitly construct the multiperiodic wave solutions of the equation. Finally, the asymptotic properties of these periodic wave solutions are strictly analyzed to reveal the relationships between periodic wave solutions and soliton solutions.
Wave Number Shocks for the Tail of Korteweg-deVries Solitary Waves in Slowly Varying Media.
1986-04-07
Kruskal and R. M. Miura (1967), Method for solving the Korteweg - deVries equation , Phys. Rev. Lett., 19: 1095-1097. [5] R. Grimshaw (1979), Slowly varying...April 7, 1986 Asymptotic solutions for the nonlinear, nonhomogeneous, Korteweg - deVries (KdV) partial differential equation <pde) with slowly varying... Korteweg and deVries . They demonstrated the existence of a permanent -1 solitary wave for nonlinear partial differential equations of shallow water theory
Saeed, R.; Shah, Asif; Noaman-ul-Haq, Muhammad
2010-10-15
The nonlinear propagation of ion-acoustic solitons in relativistic electron-positron-ion plasma comprising of Boltzmannian electrons, positrons, and relativistic thermal ions has been examined. The Korteweg-de Vries equation has been derived by reductive perturbation technique. The effect of various plasma parameters on amplitude and structure of solitary wave is investigated. The pert graphical view of the results has been presented for illustration. It is observed that increase in the relativistic streaming factor causes the soliton amplitude to thrive and its width shrinks. The soliton amplitude and width decline as the ion to electron temperature ratio is increased. The increase in positron concentration results in reduction of soliton amplitude. The soliton amplitude enhances as the electron to positron temperature ratio is increased. Our results may have relevance in the understanding of astrophysical plasmas.
NASA Astrophysics Data System (ADS)
Saeed, R.; Shah, Asif; Noaman-Ul-Haq, Muhammad
2010-10-01
The nonlinear propagation of ion-acoustic solitons in relativistic electron-positron-ion plasma comprising of Boltzmannian electrons, positrons, and relativistic thermal ions has been examined. The Korteweg-de Vries equation has been derived by reductive perturbation technique. The effect of various plasma parameters on amplitude and structure of solitary wave is investigated. The pert graphical view of the results has been presented for illustration. It is observed that increase in the relativistic streaming factor causes the soliton amplitude to thrive and its width shrinks. The soliton amplitude and width decline as the ion to electron temperature ratio is increased. The increase in positron concentration results in reduction of soliton amplitude. The soliton amplitude enhances as the electron to positron temperature ratio is increased. Our results may have relevance in the understanding of astrophysical plasmas.
Restuccia, A.; Sotomayor, A.
2013-11-15
A supersymmetric breaking procedure for N= 1 super Korteweg-de Vries (KdV), using a Clifford algebra, is implemented. Dirac's method for the determination of constraints is used to obtain the Hamiltonian structure, via a Lagrangian, for the resulting solitonic system of coupled KdV type system. It is shown that the Hamiltonian obtained by this procedure is bounded from below and in that sense represents a model which is physically admissible.
NASA Astrophysics Data System (ADS)
Zayed, Elsayed M. E.; Abdelaziz, Mahmoud A. M.
2010-12-01
In this article, a generalized (Ǵ/G)-expansion method is used to find exact travelling wave solutions of the Burgers equation and the Korteweg-de Vries (KdV) equation with variable coefficients. As a result, hyperbolic, trigonometric, and rational function solutions with parameters are obtained. When these parameters are taking special values, the solitary wave solutions are derived from the hyperbolic function solution. It is shown that the proposed method is direct, effective, and can be applied to many other nonlinear evolution equations in mathematical physics.
NASA Astrophysics Data System (ADS)
Cuesta, C. M.; Achleitner, F.
2017-01-01
We add a theorem to F. Achleitner, C.M. Cuesta and S. Hittmeir (2014) [1]. In that paper we studied travelling wave solutions of a Korteweg-de Vries-Burgers type equation with a non-local diffusion term. In particular, the proof of existence and uniqueness of these waves relies on the assumption that the exponentially decaying functions are the only bounded solutions of the linearised equation. In this addendum we prove this assumption and thus close the existence and uniqueness proof of travelling wave solutions.
NASA Astrophysics Data System (ADS)
Cheng, Wen-Guang; Qiu, De-Qin; Yu, Bo
2017-06-01
This paper is concerned with the fifth-order modified Korteweg-de Vries (fmKdV) equation. It is proved that the fmKdV equation is consistent Riccati expansion (CRE) solvable. Three special form of soliton-cnoidal wave interaction solutions are discussed analytically and shown graphically. Furthermore, based on the consistent tanh expansion (CTE) method, the nonlocal symmetry related to the consistent tanh expansion (CTE) is investigated, we also give the relationship between this kind of nonlocal symmetry and the residual symmetry which can be obtained with the truncated Painlevé method. We further study the spectral function symmetry and derive the Lax pair of the fmKdV equation. The residual symmetry can be localized to the Lie point symmetry of an enlarged system and the corresponding finite transformation group is computed. Supported by National Natural Science Foundation of China under Grant No. 11505090, and Research Award Foundation for Outstanding Young Scientists of Shandong Province under Grant No. BS2015SF009
Xing, Qiuxia; Wang, Lihong; Mihalache, Dumitru; Porsezian, Kuppuswamy; He, Jingsong
2017-05-01
In this paper, we consider the real modified Korteweg-de Vries (mKdV) equation and construct a special kind of breather solution, which can be obtained by taking the limit λj → λ1 of the Lax pair eigenvalues used in the n-fold Darboux transformation that generates the order-n periodic solution from a constant seed solution. Further, this special kind of breather solution of order n can be used to generate the order-n rational solution by taking the limit λ1 → λ0, where λ0 is a special eigenvalue associated with the eigenfunction ϕ of the Lax pair of the mKdV equation. This eigenvalue λ0, for which ϕ(λ0)=0, corresponds to the limit of infinite period of the periodic solution. Our analytical and numerical results show the effective mechanism of generation of higher-order rational solutions of the mKdV equation from the double eigenvalue degeneration process of multi-periodic solutions.
NASA Astrophysics Data System (ADS)
Tariq, Kalim Ul-Haq; Seadawy, A. R.
The Boussinesq equation with dual dispersion and modified Korteweg-de Vries-Kadomtsev-Petviashvili equations describe weakly dispersive and small amplitude waves propagating in a quasi three-dimensional media. In this article, we study the analytical Bright-Dark solitary wave solutions for (3 + 1)-dimensional Breaking soliton equation, Boussinesq equation with dual dispersion and modified Korteweg-de Vries-Kadomtsev-Petviashvili equation have been extracted. These results hold numerous travelling wave solutions that are of key importance in elucidating some physical circumstance. The technique can also be functional to other sorts of nonlinear evolution equations in contemporary areas of research.
Analysis of the small dispersion limit of a non-integrable generalized Korteweg-de Vries equation
NASA Astrophysics Data System (ADS)
Zakeri, Gholam-Ali; Yomba, Emmanuel
2013-08-01
A generalized non-integrable Korteweg-de Vries (KdV) equation is investigated for the qualitative behavior of its solutions with a small dispersion limit. We obtained two reduced ordinary differential equations using a similarity analysis and discussed the solutions of generalized KdV (gKdV) by employing singular perturbation and asymptotic methods. We found a new closed form solution and provided various approximate solutions. We have shown that for sech-type initial value data the cumulative primitive function of the gKdV solution converges point-wise as the coefficient of the dispersive term goes to zero. Our numerical experiments provide strong evidence that for each fixed time, the solutions of gKdV are bounded by well-defined envelopes as the coefficient of the dispersion term goes to zero. We have shown that for a higher order of nonlinearity, the soliton becomes shaper, with a larger amplitude, but remains bounded. Comparatively, for a smaller coefficient of the dispersion term, its base gets smaller and the soliton becomes narrower, but the amplitude of the soliton remains the same.
NASA Astrophysics Data System (ADS)
Matsutani, Shigeki
Recently there have been several studies of a nonrelativistic elastic rod in R2 whose dynamics is governed by the modified Korteweg-de Vries (MKdV) equation. Goldstein and Petrich found the MKdV hierarchy through its dynamics [Phys. Rev. Lett. 69, 555 (1992).] In this article, we will show the physical meaning of the Hirota bilinear form along the lines of the elastica problem after we formally complexify its arc length.
NASA Astrophysics Data System (ADS)
Tseluiko, D.; Papageorgiou, D. T.
2010-07-01
The gravity-driven flow of a liquid film down a vertical flat plate in the presence of an electric field acting in a direction perpendicular to the wall is investigated. The film is assumed to be a perfect conductor, and the bounding region of air above the film is taken to be a perfect dielectric. A strongly nonlinear long-wave evolution equation is developed for flow parameters near the critical instability conditions. The equation retains terms up to second order in the slenderness parameter in order to incorporate the effects of shear-induced growth, short-wave damping due to surface tension, electric stress effects, and dispersive effects. In the additional asymptotic limit of small but finite interfacial perturbations the dynamics are shown to be governed by a Kuramoto-Sivashinsky (KS) equation with Korteweg-de Vries dispersion, also known as the Kawahara or the generalized Kuramoto-Sivashinsky (gKS) equation, which also includes a nonlocal energy growth term that arises from the electrostatics. Extensive numerical experiments are carried out to characterize solutions to this equation. Using perturbation theory and numerical solutions, it is shown that the electric field alters the far-field decay characteristics of bound states of the gKS equation from exponential to algebraic behavior. In addition, it is demonstrated numerically that chaotic solutions of the KS equation that are regularized into traveling-wave pulses when sufficient dispersion is added can in turn become chaotic by applying a sufficiently strong electric field. It is suggested, therefore, that electric fields can be utilized to enhance interfacial turbulence and in turn increase heat or mass transfer in applications. A physical example involving electrified falling film flows of ethelene glycol fluids is furnished and shows that the theory is within reach of experiments.
NASA Astrophysics Data System (ADS)
Hosen, B.; Amina, M.; Mamun, A. A.; Hossen, M. R.
2016-12-01
The nonlinear properties of ion-acoustic (IA) waves are investigated in a relativistically degenerate magnetized quantum plasma, whose constituents are non-degenerate inertial ions, degenerate electrons and immobile positively-charged heavy elements. For nonlinear studies, the well-known reductive perturbation technique is employed to derive the Korteweg-de Vries-Burger equation in the presence of relativistically degenerate electrons. Numerically, the amplitude, width, and phase speed are shown to be associated with the localized IA solitons, and shocks are shown to be significantly influenced by the various intrinsic parameters relevant to our model. The solitary and the shock wave properties have been to be influenced in the non-relativistic, as well as the ultrarelativistic, limits. The effects of the external magnetic field and the obliqueness are found to change the basic properties of IA waves significantly. The present analysis can be useful in understanding the collective process in dense astrophysical environments, like there of non-rotating white dwarfs, neutron stars, etc.
El-Tantawy, S. A.; Moslem, W. M.
2014-05-15
Solitons (small-amplitude long-lived waves) collision and rogue waves (large-amplitude short-lived waves) in non-Maxwellian electron-positron-ion plasma have been investigated. For the solitons collision, the extended Poincaré-Lighthill-Kuo perturbation method is used to derive the coupled Korteweg-de Vries (KdV) equations with the quadratic nonlinearities and their corresponding phase shifts. The calculations reveal that both positive and negative polarity solitons can propagate in the present model. At critical value of plasma parameters, the coefficients of the quadratic nonlinearities disappear. Therefore, the coupled modified KdV (mKdV) equations with cubic nonlinearities and their corresponding phase shifts have been derived. The effects of the electron-to-positron temperature ratio, the ion-to-electron temperature ratio, the positron-to-ion concentration, and the nonextensive parameter on the colliding solitons profiles and their corresponding phase shifts are examined. Moreover, generation of ion-acoustic rogue waves from small-amplitude initial perturbations in plasmas is studied in the framework of the mKdV equation. The properties of the ion-acoustic rogue waves are examined within a nonlinear Schrödinger equation (NLSE) that has been derived from the mKdV equation. The dependence of the rogue wave profile on the relevant physical parameters has been investigated. Furthermore, it is found that the NLSE that has been derived from the KdV equation cannot support the propagation of rogue waves.
Exact Solutions for a Coupled Korteweg-de Vries System
NASA Astrophysics Data System (ADS)
Zuo, Da-Wei; Jia, Hui-Xian
2016-11-01
Korteweg-de Vries (KdV)-type equation can be used to characterise the dynamic behaviours of the shallow water waves and interfacial waves in the two-layer fluid with gradually varying depth. In this article, by virtue of the bilinear forms, rational solutions and three kind shapes (soliton-like, kink and bell, anti-bell, and bell shapes) for the Nth-order soliton-like solutions of a coupled KdV system are derived. Propagation and interaction of the solitons are analyzed: (1) Potential u shows three kind of shapes (soliton-like, kink, and anti-bell shapes); Potential v exhibits two type of shapes (soliton-like and bell shapes); (2) Interaction of the potentials u and v both display the fusion phenomena.
NASA Astrophysics Data System (ADS)
Li, Li-Li; Tian, Bo; Zhang, Chun-Yi; Zhang, Hai-Qiang; Li, Juan; Xu, Tao
In this paper, a nonisospectral and variable-coefficient Korteweg-de Vries equation is investigated based on the ideas of the variable-coefficient balancing-act method and Hirota method. Via symbolic computation, we obtain the analytic N-soliton solutions, variable-coefficient bilinear form, auto-Bäcklund transformations (in both the bilinear form and Lax pair form), Lax pair and nonlinear superposition formula for such an equation in explicit form. Moreover, some figures are plotted to analyze the effects of the variable coefficients on the stabilities and propagation characteristics of the solitonic waves.
Compactons in PT-symmetric generalized Korteweg-de Vries equations
Saxena, Avadh B; Mihaila, Bogdan; Bender, Carl M; Cooper, Fred; Khare, Avinash
2008-01-01
In an earlier paper Cooper, Shepard, and Sodano introduced a generalized KdV equation that can exhibit the kinds of compacton solitary waves that were first seen in equations studied by Rosenau and Hyman. This paper considers the PT-symmetric extensions of the equations examined by Cooper, Shepard, and Sodano. From the scaling properties of the PT-symmetric equations a general theorem relating the energy, momentum, and velocity of any solitary-wave solution of the generalized KdV equation is derived, and it is shown that the velocity of the solitons is determined by their amplitude, width, and momentum.
Asymptotic behavior of solutions of the Korteweg-de Vries equation
Buslaev, V.S.
1986-09-01
For the KdV equation a complete asymptotic expansion of the ''dispersive tail'' for large times is described, and generalized wave operators are introduced. The asymptotics for large times of the spectral Schrodinger equation with a potential of the type of a solution of the KdV equation is studied. It is shown that the KdV equation is connected in a specific manner with the structure of the asymptotics of solutions of the spectral equation. As a corollary, known explicit formulas for the leading terms of the asymptotics of solutions of the KdV equation in terms of spectral data corresponding to the initial conditions are obtained. A plan for justifying the results listed is outlined.
Asymptotic behavior of solutions of the Korteweg-de Vries equation for large times
Buslaev, V.S.; Sukhanov, V.V.
1986-09-10
For the KdV equation a complete asymptotic expansion of the dispersive tail for large times is described, and generalized wave operators are introduced. The asymptotics for large times of the spectral Schroedinger equation with a potential of the type of a solution of the KdV equation is studied. It is shown that the KdV equation is connected in a specific manner with the structure of the asymptotics of solutions of the spectral equation. As a corollary, known explicit formulas for the leading terms of the asymptotics of solutions of the KdV equation in terms of spectral data corresponding to the initial conditions are obtained. A plan for justifying the results listed is outlined.
Long-time asymptotic behavior of the solutions of the Korteweg-De Vries equations
Buslaev, V.S.; Sukhanov, V.V.
1987-05-20
The complete asymptotic expansion of the dispersion tail in the long-time limit is described for the KdV equation and generalized wave operators are introduced. The long-time asymptotic behavior of the Schroedinger spectral equation is studied assuming a potential of the type of the KdV solution. It is shown that the KdV equation is specifically related with the asymptotic structure of the solutions of the spectral equation. As a corollary, they derive the well-known explicit formulas for the leading asymptotic terms of the KdV solutions in terms of the spectral values corresponding to the initial conditions. A sketch of a proof for the various results is suggested.
Korteweg-de Vries-Kuramoto-Sivashinsky filters in biomedical image processing
NASA Astrophysics Data System (ADS)
Arango, Juan C.
2015-09-01
The Kuramoto- Sivashinsky operator is applied to the two-dimensional solution of the Korteweg-de Vries equation and the resulting filter is applied via convolution to image processing. The full procedure is implemented using an algorithm: prototyped with the Maple package named Image Tools. Some experiments were performed using biomedical images, infrared images obtained with smartphones and images generated by photon diffusion. The results from these experiments show that the Kuramoto-Sivashinsky-Korteweg-de Vries Filters are excellent tools for enhancement of images with interesting applications in image processing at general and biomedical image processing in particular. It is expected that the incorporation of the Kuramoto-Sivashinsky-Korteweg-de Vries Filters in standard programs for image processing will led to important improvements in various fields of optical engineering.
NASA Astrophysics Data System (ADS)
Yang, Xiao; Han, Jiayan
2017-07-01
A generalised Kaup-Newell (gKN) hierarchy is introduced, which starts with a system of first-order ordinary differential equations and includes the Gerdjikov-Ivanov equation. By introducing an appropriate generating function, its related Hamiltonian systems and algebraic curve are given. The Hamiltonian systems are proved to be integrable, then the gKN hierarchy is solved by Hamiltonian flows. The algebraic curve is provided with suitable genus, then based on the trace formula and Riemann-Jacobi inversion theorem, finite genus solutions of the gKN hierarchy are obtained. Besides, two 2+1 dimensional modified Korteweg-de Vries (mKdV) equations are also solved.
NASA Astrophysics Data System (ADS)
Zhang, Yingnan; Tam, Hon-Wah; Hu, Xingbiao
2014-01-01
This paper presents a new integrable discretization of the Korteweg-de Vries (KdV) equation. Different from other discrete analogues, we discretize the variable ‘time’ and obtain an integrable differential-difference system. This system has the original KdV equation as a standard limit when the step size tends to zero. The main idea is based on Hirota’s bilinear method and Bäcklund transformation and can be applied to other integrable systems. By applying the Fourier pseudospectral method to the space variable, we derive a new numerical scheme for the KdV equation. Numerical results are found to agree with the exact solution and the first five conservation quantities are preserved quite well.
NASA Astrophysics Data System (ADS)
Deng, Guo; Biondini, Gino; Trillo, Stefano
2016-10-01
We study the small dispersion limit of the Korteweg-de Vries (KdV) equation with periodic boundary conditions and we apply the results to the Zabusky-Kruskal experiment. In particular, we employ a WKB approximation for the solution of the scattering problem for the KdV equation [i.e., the time-independent Schrödinger equation] to obtain an asymptotic expression for the trace of the monodromy matrix and thereby of the spectrum of the problem. We then perform a detailed analysis of the structure of said spectrum (i.e., band widths, gap widths and relative band widths) as a function of the dispersion smallness parameter ɛ. We then formulate explicit approximations for the number of solitons and corresponding soliton amplitudes as a function of ɛ. Finally, by performing an appropriate rescaling, we compare our results to those in the famous Zabusky and Kruskal's paper, showing very good agreement with the numerical results.
Symmetry breaking in linearly coupled Korteweg-de Vries systems.
Espinosa-Cerón, A; Malomed, B A; Fujioka, J; Rodríguez, R F
2012-09-01
We consider solitons in a system of linearly coupled Korteweg-de Vries (KdV) equations, which model two-layer settings in various physical media. We demonstrate that traveling symmetric solitons with identical components are stable at velocities lower than a certain threshold value. Above the threshold, which is found exactly, the symmetric modes are unstable against spontaneous symmetry breaking, which gives rise to stable asymmetric solitons. The shape of the asymmetric solitons is found by means of a variational approximation and in the numerical form. Simulations of the evolution of an unstable symmetric soliton sometimes produce its breakup into two different asymmetric modes. Collisions between moving stable solitons, symmetric and asymmetric ones, are studied numerically, featuring noteworthy features. In particular, collisions between asymmetric solitons with identical polarities are always elastic, while in the case of opposite polarities the collision leads to a switch of the polarities of both solitons. Three-soliton collisions are studied too, featuring quite complex interaction scenarios.
NASA Astrophysics Data System (ADS)
Li, He; Gao, Yi-Tian; Liu, Li-Cai
2015-12-01
The Korteweg-de Vries (KdV)-type equations have been seen in fluid mechanics, plasma physics and lattice dynamics, etc. This paper will address the bilinearization problem for some higher-order KdV equations. Based on the relationship between the bilinear method and Bell-polynomial scheme, with introducing an auxiliary independent variable, we will present the general bilinear forms. By virtue of the symbolic computation, one- and two-soliton solutions are derived. Supported by the National Natural Science Foundation of China under Grant No. 11272023, the Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), and by the Fundamental Research Funds for the Central Universities of China under Grant No. 2011BUPTYB02
Deift, P; Venakides, S; Zhou, X
1998-01-20
This paper extends the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou in a critical new way. We present, in particular, an algorithm, to obtain the support of the Riemann-Hilbert problem for leading asymptotics. Applying this extended method to small dispersion KdV (Korteweg-de Vries) equation, we (i) recover the variational formulation of P. D. Lax and C. D. Levermore [(1979) Proc. Natl. Acad. Sci. USA76, 3602-3606] for the weak limit of the solution, (ii) derive, without using an ansatz, the hyperelliptic asymptotic solution of S. Venakides that describes the oscillations; and (iii) are now able to compute the phase shifts, integrating the modulation equations exactly. The procedure of this paper is a version of fully nonlinear geometrical optics for integrable systems. With some additional analysis the theory can provide rigorous error estimates between the solution and its computed asymptotic expression.
Deift, P.; Venakides, S.; Zhou, X.
1998-01-01
This paper extends the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou in a critical new way. We present, in particular, an algorithm, to obtain the support of the Riemann-Hilbert problem for leading asymptotics. Applying this extended method to small dispersion KdV (Korteweg-de Vries) equation, we (i) recover the variational formulation of P. D. Lax and C. D. Levermore [(1979) Proc. Natl. Acad. Sci. USA 76, 3602–3606] for the weak limit of the solution, (ii) derive, without using an ansatz, the hyperelliptic asymptotic solution of S. Venakides that describes the oscillations; and (iii) are now able to compute the phase shifts, integrating the modulation equations exactly. The procedure of this paper is a version of fully nonlinear geometrical optics for integrable systems. With some additional analysis the theory can provide rigorous error estimates between the solution and its computed asymptotic expression. PMID:11038618
NASA Astrophysics Data System (ADS)
Shen, Y.; Kevrekidis, P. G.; Sen, S.; Hoffman, A.
2014-08-01
Our aim in the present work is to develop approximations for the collisional dynamics of traveling waves in the context of granular chains in the presence of precompression. To that effect, we aim to quantify approximations of the relevant Hertzian FPU-type lattice through both the Korteweg-de Vries (KdV) equation and the Toda lattice. Using the availability in such settings of both one-soliton and two-soliton solutions in explicit analytical form, we initialize such coherent structures in the granular chain and observe the proximity of the resulting evolution to the underlying integrable (KdV or Toda) model. While the KdV offers the possibility to accurately capture collisions of solitary waves propagating in the same direction, the Toda lattice enables capturing both copropagating and counterpropagating soliton collisions. The error in the approximation is quantified numerically and connections to bounds established in the mathematical literature are also given.
Berbri, Abderrezak; Tribeche, Mouloud
2009-05-15
A weakly nonlinear analysis is carried out to derive a Korteweg-de Vries Burgers-like equation for small but finite amplitude dust ion-acoustic (DIA) waves in a charge varying dusty plasma with non thermally distributed electrons. The correct expression for the nonthermal electron charging current is used. Interestingly, it may be noted that due to electron nonthermality and finite equilibrium ion streaming velocity, the present dusty plasma model can admit compressive as well as rarefactive DIA solitary waves. Furthermore, there may exist DIA shocks which have either monotonic or oscillatory behavior and the properties of which depend sensitively on the number of fast nonthermal electrons. Our results should be useful to understand the properties of localized DIA waves that may occur in space dusty plasmas.
Shen, Y; Kevrekidis, P G; Sen, S; Hoffman, A
2014-08-01
Our aim in the present work is to develop approximations for the collisional dynamics of traveling waves in the context of granular chains in the presence of precompression. To that effect, we aim to quantify approximations of the relevant Hertzian FPU-type lattice through both the Korteweg-de Vries (KdV) equation and the Toda lattice. Using the availability in such settings of both one-soliton and two-soliton solutions in explicit analytical form, we initialize such coherent structures in the granular chain and observe the proximity of the resulting evolution to the underlying integrable (KdV or Toda) model. While the KdV offers the possibility to accurately capture collisions of solitary waves propagating in the same direction, the Toda lattice enables capturing both copropagating and counterpropagating soliton collisions. The error in the approximation is quantified numerically and connections to bounds established in the mathematical literature are also given.
NASA Astrophysics Data System (ADS)
Feng, Zhaosheng
Many physical phenomena can be described by nonlinear models. The last few decades have seen an enormous growth of the applicability of nonlinear models and of the development of related nonlinear concepts. This has been driven by modern computer power as well as by the discovery of new mathematical techniques, which include two contrasting themes: (i) the theory of dynamical systems, most popularly associated with the study of chaos, and (ii) the theory of integrable systems associated, among other things, with the study of solitons. In this dissertation, we study two nonlinear models. One is the 1-dimensional vibrating string satisfying wtt - wxx = 0 with van der Pol boundary conditions. We formulate the problem into an equivalent first order Hyperbolic system, and use the method of characteristics to derive a nonlinear reflection relation caused by the nonlinear boundary conditions. Thus, the problem is reduced to the discrete iteration problem of the type un+1 = F( un). Periodic solutions are investigated, an invariant interval for the Abel equation is studied, and numerical simulations and visualizations with different coefficients are illustrated. The other model is the Korteweg-de Vries-Burgers (KdVB) equation. In this dissertation, we proposed two new approaches: One is what we currently call First Integral Method, which is based on the ring theory of commutative algebra. Applying the Hilbert-Nullstellensatz, we reduce the KdVB equation to a first-order integrable ordinary differential equation. The other approach is called the Coordinate Transformation Method, which involves a series of variable transformations. Some new results on the traveling wave solution are established by using these two methods, which not only are more general than the existing ones in the previous literature, but also indicate that some corresponding solutions presented in the literature contain errors. We clarify the errors and instead give a refined result.
NASA Astrophysics Data System (ADS)
Ganguly, A.; Das, A.
2014-11-01
We consider one-dimensional stationary position-dependent effective mass quantum model and derive a generalized Korteweg-de Vries (KdV) equation in (1+1) dimension through Lax pair formulation, one being the effective mass Schrödinger operator and the other being the time-evolution of wave functions. We obtain an infinite number of conserved quantities for the generated nonlinear equation and explicitly show that the new generalized KdV equation is an integrable system. Inverse scattering transform method is applied to obtain general solution of the nonlinear equation, and then N-soliton solution is derived for reflectionless potentials. Finally, a special choice has been made for the variable mass function to get mass-deformed soliton solution. The influence of position and time-dependence of mass and also of the different representations of kinetic energy operator on the nature of such solitons is investigated in detail. The remarkable features of such solitons are demonstrated in several interesting figures and are contrasted with the conventional KdV-soliton associated with constant-mass quantum model.
Ganguly, A. E-mail: aganguly@maths.iitkgp.ernet.in; Das, A.
2014-11-15
We consider one-dimensional stationary position-dependent effective mass quantum model and derive a generalized Korteweg-de Vries (KdV) equation in (1+1) dimension through Lax pair formulation, one being the effective mass Schrödinger operator and the other being the time-evolution of wave functions. We obtain an infinite number of conserved quantities for the generated nonlinear equation and explicitly show that the new generalized KdV equation is an integrable system. Inverse scattering transform method is applied to obtain general solution of the nonlinear equation, and then N-soliton solution is derived for reflectionless potentials. Finally, a special choice has been made for the variable mass function to get mass-deformed soliton solution. The influence of position and time-dependence of mass and also of the different representations of kinetic energy operator on the nature of such solitons is investigated in detail. The remarkable features of such solitons are demonstrated in several interesting figures and are contrasted with the conventional KdV-soliton associated with constant-mass quantum model.
NASA Astrophysics Data System (ADS)
O'Driscoll, Kieran; Levine, Murray
2017-09-01
Numerical solutions of the Korteweg-de Vries (KdV) and extended Korteweg-de Vries (eKdV) equations are used to model the transformation of a sinusoidal internal tide as it propagates across the continental shelf. The ocean is idealized as being a two-layer fluid, justified by the fact that most of the oceanic internal wave signal is contained in the gravest mode. The model accounts for nonlinear and dispersive effects but neglects friction, rotation and mean shear. The KdV model is run for a number of idealized stratifications and unique realistic topographies to study the role of the nonlinear and dispersive effects. In all model solutions the internal tide steepens forming a sharp front from which a packet of nonlinear solitary-like waves evolve. Comparisons between KdV and eKdV solutions are made. The model results for realistic topography and stratification are compared with observations made at moorings off Massachusetts in the Middle Atlantic Bight. Some features of the observations compare well with the model. The leading face of the internal tide steepens to form a shock-like front, while nonlinear high-frequency waves evolve shortly after the appearance of the jump. Although not rank ordered, the wave of maximum amplitude is always close to the jump. Some features of the observations are not found in the model. Nonlinear waves can be very widely spaced and persist over a tidal period.
NASA Astrophysics Data System (ADS)
Sun, Wen-Rong; Shan, Wen-Rui; Jiang, Yan; Wang, Pan; Tian, Bo
2015-02-01
The fifth-order Korteweg-de Vries (KdV) equation works as a model for the shallow water waves with surface tension. Through symbolic computation, binary Bell-polynomial approach and auxiliary independent variable, the bilinear forms, N-soliton solutions, two different Bell-polynomial-type Bäcklund transformations, Lax pair and infinite conservation laws are obtained. Characteristic-line method is applied to discuss the effects of linear wave speed c as well as length scales τ and γ on the soliton amplitudes and velocities. Increase of τ, c and γ can lead to the increase of the soliton velocity. Soliton amplitude increases with the increase of τ. The larger-amplitude soliton is seen to move faster and then to overtake the smaller one. After the collision, the solitons keep their original shapes and velocities invariant except for the phase shift. Graphic analysis on the two and three-soliton solutions indicates that the overtaking collisions between/among the solitons are all elastic.
NASA Astrophysics Data System (ADS)
Saeed, R.; Shah, Asif
2010-03-01
The nonlinear propagation of ion acoustic waves in electron-positron-ion plasma comprising of Boltzmannian electrons, positrons, and relativistic thermal ions has been examined. The Korteweg-de Vries-Burger equation has been derived by reductive perturbation technique, and its shock like solution is determined analytically through tangent hyperbolic method. The effect of various plasma parameters on strength and structure of shock wave is investigated. The pert graphical view of the results has been presented for illustration. It is observed that strength and steepness of the shock wave enervate with an increase in the ion temperature, relativistic streaming factor, positron concentrations, electron temperature and they accrue with an increase in coefficient of kinematic viscosity. The convective, dispersive, and dissipative properties of the plasma are also discussed. It is determined that the electron temperature has remarkable influence on the propagation and structure of nonlinear wave in such relativistic plasmas. The numerical analysis has been done based on the typical numerical data from a pulsar magnetosphere.
Saeed, R.; Shah, Asif
2010-03-15
The nonlinear propagation of ion acoustic waves in electron-positron-ion plasma comprising of Boltzmannian electrons, positrons, and relativistic thermal ions has been examined. The Korteweg-de Vries-Burger equation has been derived by reductive perturbation technique, and its shock like solution is determined analytically through tangent hyperbolic method. The effect of various plasma parameters on strength and structure of shock wave is investigated. The pert graphical view of the results has been presented for illustration. It is observed that strength and steepness of the shock wave enervate with an increase in the ion temperature, relativistic streaming factor, positron concentrations, electron temperature and they accrue with an increase in coefficient of kinematic viscosity. The convective, dispersive, and dissipative properties of the plasma are also discussed. It is determined that the electron temperature has remarkable influence on the propagation and structure of nonlinear wave in such relativistic plasmas. The numerical analysis has been done based on the typical numerical data from a pulsar magnetosphere.
NASA Astrophysics Data System (ADS)
Melnikov, Andrey
2017-05-01
To study Korteweg-de Vries (KdV) equation, whose solution is based on the evolution of the Sturm-Liouville (SL) operator, we have recently developed scattering theory of the SL operator for an arbitrary analytic potential on the line. Here, we create a basis for solutions of the KdV of a different class, considering continuously differentiable potentials using the Gelfand-Levitan theory on the half line. The solution of the spectral problem for this operator is achieved by constructing a special object, invented by Livšic in the late 1970s, and called vessel. The central object of a vessel is a compact perturbation of a self-adjoint operator for which we develop a canonical form. Evolving the constructed vessel, we solve the KdV equation on the half line, coinciding with the given potential for t = 0. It is also shown that the initial value on the t-axis, or equivalently for x = 0, is prescribed by the choice of the spectral parameters, but can be perturbed using an "orthogonal" to the problem measure. The results, presented in this work, are following: (1) We address the KdV equation with initial values, satisfying Gelfand-Levitan theory assumptions, providing a detailed formula of the initial conditions for the x = 0, (2) we show that Nonlinear Shrodinger, canonical systems, and many more equations can be solved using theory of vessels, analogously to the Zacharov-Shabbath scheme, (3) we present a generalized inverse scattering theory on a line for potentials with singularities using prevessels, and (4) we present the tau function and its role in the solution of the problem.
Dispersion management for solitons in a Korteweg-de Vries system.
Clarke, Simon; Malomed, Boris A.; Grimshaw, Roger
2002-03-01
The existence of "dispersion-managed solitons," i.e., stable pulsating solitary-wave solutions to the nonlinear Schrodinger equation with periodically modulated and sign-variable dispersion is now well known in nonlinear optics. Our purpose here is to investigate whether similar structures exist for other well-known nonlinear wave models. Hence, here we consider as a basic model the variable-coefficient Korteweg-de Vries equation; this has the form of a Korteweg-de Vries equation with a periodically varying third-order dispersion coefficient, that can take both positive and negative values. More generally, this model may be extended to include fifth-order dispersion. Such models may describe, for instance, periodically modulated waveguides for long gravity-capillary waves. We develop an analytical approximation for solitary waves in the weakly nonlinear case, from which it is possible to obtain a reduction to a relatively simple integral equation, which is readily solved numerically. Then, we describe some systematic direct simulations of the full equation, which use the soliton shape produced by the integral equation as an initial condition. These simulations reveal regions of stable and unstable pulsating solitary waves in the corresponding parametric space. Finally, we consider the effects of fifth-order dispersion. (c) 2002 American Institute of Physics.
Superregular breathers in a complex modified Korteweg-de Vries system
NASA Astrophysics Data System (ADS)
Liu, Chong; Ren, Yang; Yang, Zhan-Ying; Yang, Wen-Li
2017-08-01
We study superregular (SR) breathers (i.e., the quasi-Akhmediev breather collision with a certain phase shift) in a complex modified Korteweg-de Vries equation. We demonstrate that such SR waves can exhibit intriguing nonlinear structures, including the half-transition and full-suppression modes, which have no analogues in the standard nonlinear Schrödinger equation. In contrast to the standard SR breather formed by pairs of quasi-Akhmediev breathers, the half-transition mode describes a mix of quasi-Akhmediev and quasi-periodic waves, whereas the full-suppression mode shows a non-amplifying nonlinear dynamics of localized small perturbations associated with the vanishing growth rate of modulation instability. Interestingly, we show analytically and numerically that these different SR modes can be evolved from an identical localized small perturbation. In particular, our results demonstrate an excellent compatibility relation between SR modes and the linear stability analysis.
NASA Astrophysics Data System (ADS)
Selima, Ehab S.; Yao, Xiaohua; Wazwaz, Abdul-Majid
2017-06-01
In this research, the surface waves of a horizontal fluid layer open to air under gravity field and vertical temperature gradient effects are studied. The governing equations of this model are reformulated and converted to a nonlinear evolution equation, the perturbed Korteweg-de Vries (pKdV) equation. We investigate the latter equation, which includes dispersion, diffusion, and instability effects, in order to examine the evolution of long surface waves in a convective fluid. Dispersion relation of the pKdV equation and its properties are discussed. The Painlevé analysis is applied not only to check the integrability of the pKdV equation but also to establish the Bäcklund transformation form. In addition, traveling wave solutions and a general form of the multiple-soliton solutions of the pKdV equation are obtained via Bäcklund transformation, the simplest equation method using Bernoulli, Riccati, and Burgers' equations as simplest equations, and the factorization method.
Korteweg-de Vries solitons on electrified liquid jets
NASA Astrophysics Data System (ADS)
Wang, Qiming; Papageorgiou, Demetrios T.; Vanden-Broeck, Jean-Marc
2015-06-01
The propagation of axisymmetric waves on the surface of a liquid jet under the action of a radial electric field is considered. The jet is assumed to be inviscid and perfectly conducting, and a field is set up by placing the jet concentrically inside a perfectly cylindrical tube whose wall is maintained at a constant potential. A nontrivial interaction arises between the hydrodynamics and the electric field in the annulus, resulting in the formation of electrocapillary waves. The main objective of the present study is to describe nonlinear aspects of such axisymmetric waves in the weakly nonlinear regime, which is valid for long waves relative to the undisturbed jet radius. This is found to be possible if two conditions hold: the outer electrode radius is not too small, and the applied electric field is sufficiently strong. Under these conditions long waves are shown to be dispersive and a weakly nonlinear theory can be developed to describe the evolution of the disturbances. The canonical system that arises is the Kortweg-de Vries equation with coefficients that vary as the electric field and the electrode radius are varied. Interestingly, the coefficient of the highest-order third derivative term does not change sign and remains strictly positive, whereas the coefficient α of the nonlinear term can change sign for certain values of the parameters. This finding implies that solitary electrocapillary waves are possible; there are waves of elevation for α >0 and of depression for α <0 . Regions in parameter space are identified where such waves are found.
Nonautonomous analysis of steady Korteweg-de Vries waves under nonlocalised forcing
NASA Astrophysics Data System (ADS)
Balasuriya, Sanjeeva; Binder, Benjamin J.
2014-10-01
Recently developed nonautonomous dynamical systems theory is applied to quantify the effect of bottom topography variation on steady surface waves governed by the Korteweg-de Vries (KdV) equation. Arbitrary (but small) nonlocalised bottom topographies are amenable to this method. Two classes of free surface solutions-hyperbolic and homoclinic solutions of the associated augmented dynamical system-are characterised. The first of these corresponds to near-uniform free-surface flows for which explicit formulæ are developed for a range of topographies. The second corresponds to solitary waves on the free surface, and a method for determining their number is developed. Formulæ for the shape of these solitary waves are also obtained. Theoretical free-surface profiles are verified using numerical KdV solutions, and excellent agreement is obtained.
Electrostatic Korteweg-deVries solitary waves in a plasma with Kappa-distributed electrons
Choi, C.-R.; Min, K.-W.; Rhee, T.-N.
2011-09-15
The Korteweg-deVries (KdV) equation that describes the evolution of nonlinear ion-acoustic solitary waves in plasmas with Kappa-distributed electrons is derived by using a reductive perturbation method in the small amplitude limit. We identified a dip-type (negative) electrostatic KdV solitary wave, in addition to the hump-type solution reported previously. The two types of solitary waves occupy different domains on the {kappa} (Kappa index)-V (propagation velocity) plane, separated by a curve corresponding to singular solutions with infinite amplitudes. For a given Kappa value, the dip-type solitary wave propagates faster than the hump-type. It was also found that the hump-type solitary waves cannot propagate faster than V = 1.32.
Anomalous autoresonance threshold for chirped-driven Korteweg-de-Vries waves.
Friedland, L; Shagalov, A G; Batalov, S V
2015-10-01
Large amplitude traveling waves of the Korteweg-de-Vries (KdV) equation can be excited and controlled by a chirped frequency driving perturbation. The process involves capturing the wave into autoresonance (a continuous nonlinear synchronization) with the drive by passage through the linear resonance in the problem. The transition to autoresonance has a sharp threshold on the driving amplitude. In all previously studied autoresonant problems the threshold was found via a weakly nonlinear theory and scaled as α(3/4),α being the driving frequency chirp rate. It is shown that this scaling is violated in a long wavelength KdV limit because of the increased role of the nonlinearity in the problem. A fully nonlinear theory describing the phenomenon and applicable to all wavelengths is developed.
1987-09-25
for the Korteweg - deVries equation . In order to understand the effects of a slowly varying medium, Luke [1] in 1966 utilized themethod of multiple... Korteweg - deVries type equations [7]. For clarity, we note that after using (4.9) and VkJ : 0 [see (7.6)] the equation for the modulated phase shift O(X,T...dispersive oscillatory waves are analyzed for Korteweg - deVries type partial differential equations with slowly varying coefficients and arbitrary
Static algebraic solitons in Korteweg-de Vries type systems and the Hirota transformation.
Burde, G I
2011-08-01
Some effects in the soliton dynamics governed by higher-order Korteweg-de Vries (KdV) type equations are discussed. This is done based on the exact explicit solutions of the equations derived in the paper. It is shown that some higher order KdV equations possessing multisoliton solutions also admit steady state solutions in terms of algebraic functions describing localized patterns. Solutions including both those static patterns and propagating KdV-like solitons are combinations of algebraic and hyperbolic functions. It is shown that the localized structures behave like static solitons upon collisions with regular moving solitons, with their shape remaining unchanged after the collision and only the position shifted. These phenomena are not revealed in common multisoliton solutions derived using inverse scattering or Hirota's method. The solutions of the higher-order KdV type equations were obtained using a method devised for obtaining soliton solutions of nonlinear evolution equations. This method can be combined with Hirota's method with a modified representation of the solution which allows the results to be extended to multisoliton solutions. The prospects for applying the methods to soliton equations not of KdV type are discussed.
NASA Astrophysics Data System (ADS)
Sun, Fu-Wei; Gao, Yi-Tian; Zhang, Chun-Yi; Xu, Xiao-Ge
We investigate a generalized variable-coefficient modified Korteweg-de Vries model with perturbed factor and external force (vc-GmKdV) describing fluid dynamics and space plasmas. In this paper, we propose an extended variable-coefficient balancing-act method (Evc-BAM), which is concise and straightforward, to obtain the generalized analytic solutions including solitary wave solution of the vc-GmKdV model with symbolic computation. Meanwhile, using the Evc-BAM, we obtain an auto-Bäcklund transformation for the vc-GmKdV model on the relevant constraint conditions of the coefficient functions. Using the given auto-Bäcklund transformation, the solutions of special equations for the vc-GmKdV model are also obtained as the variable-coefficient Korteweg-de Vries (vc-KdV) equation, the generalized KdV equation with perturbed factor and external force (GKdV), the variable-coefficient modified Korteweg-de Vries (vc-mKdV) equation, and the variable-coefficient cylindrical modified Korteweg-de Vries (vc-cmKdV) equation, respectively.
Modified Korteweg-de Vries solitons at supercritical densities in two-electron temperature plasmas
NASA Astrophysics Data System (ADS)
Verheest, Frank; Olivier, Carel P.; Hereman, Willy A.
2016-04-01
> The supercritical composition of a plasma model with cold positive ions in the presence of a two-temperature electron population is investigated, initially by a reductive perturbation approach, under the combined requirements that there be neither quadratic nor cubic nonlinearities in the evolution equation. This leads to a unique choice for the set of compositional parameters and a modified Korteweg-de Vries equation (mKdV) with a quartic nonlinear term. The conclusions about its one-soliton solution and integrability will also be valid for more complicated plasma compositions. Only three polynomial conservation laws can be obtained. The mKdV equation with quartic nonlinearity is not completely integrable, thus precluding the existence of multi-soliton solutions. Next, the full Sagdeev pseudopotential method has been applied and this allows for a detailed comparison with the reductive perturbation results. This comparison shows that the mKdV solitons have slightly larger amplitudes and widths than those obtained from the more complete Sagdeev solution and that only slightly superacoustic mKdV solitons have acceptable amplitudes and widths, in the light of the full solutions.
Dust-acoustic Korteweg-de Vries solitons in an adiabatic hot dusty plasma
Sayed, Fatema; Mamun, A. A.
2007-01-15
A rigorous theoretical investigation has been made of dust-acoustic (DA) Korteweg-de Vries (K-dV) solitons by the reductive perturbation method. An unmagnetized dusty plasma consisting of negatively charged adiabatic hot dust fluid and of Boltzmann distributed electrons and ions has been considered. It has been found that the DA K-dV solitons associated with only negative potential can exist in such a dusty plasma. It has been also found that the effects of dust fluid temperature have significantly modified the basic properties (amplitude and width) of the solitary potential structures in such a dusty plasma. The implications of these results to some space and astrophysical plasma situations are briefly discussed.
NASA Astrophysics Data System (ADS)
Aminmansoor, F.; Abbasi, H.
2015-08-01
The present paper is devoted to simulation of nonlinear disintegration of a localized perturbation into ion-acoustic solitons train in a plasma with hot electrons and cold ions. A Gaussian initial perturbation is used to model the localized perturbation. For this purpose, first, we reduce fluid system of equations to a Korteweg de-Vries equation by the following well-known assumptions. (i) On the ion-acoustic evolution time-scale, the electron velocity distribution function (EVDF) is assumed to be stationary. (ii) The calculation is restricted to small amplitude cases. Next, in order to generalize the model to finite amplitudes cases, the evolution of EVDF is included. To this end, a hybrid code is designed to simulate the case, in which electrons dynamics is governed by Vlasov equation, while cold ions dynamics is, like before, studied by the fluid equations. A comparison between the two models shows that although the fluid model is capable of demonstrating the general features of the process, to have a better insight into the relevant physics resulting from the evolution of EVDF, the use of kinetic treatment is of great importance.
Allgaier, D.E.
1986-04-07
Asymptotic solutions for the nonlinear, nonhomogeneous, Korteweg-deVries (KdV) partial differential equation with slowly varying coefficients are not, in general, uniformly valid. A uniform asymptotic expansion is obtained by finding separate expansions for different regions and matching. A KdV solitary wave propagating in slowly varying media is examined. Quasi-stationarity for the core reduces the problem to solving ordinary differential equations for that region. However, in the leading tail region, hyperbolic pde's must be solved to determine the amplitude and phase. The method of characteristics predicts triple valuedness after a caustic (penumbral or cusped) develops. Singular perturbation methods show the solution near first focusing satisfies the diffusion equation and involves either an incomplete Airy-type integral or an exponential integral similar to the Pearcey integral. Laplace's method shows that the critical points of the exponential phase satisfy the fundamental folding equation. A linear multi-phase solution is determined which does not become triple valued (break). Instead, a wave number shock develops, which separates two different solitary wave tails, and travels at the shock velocity predicted by conservation of waves. Thus, a unique uniform leading tail solution is obtained corresponding to a specified moving core (the problem is shown to be well-posed).
Aminmansoor, F.; Abbasi, H.
2015-08-15
The present paper is devoted to simulation of nonlinear disintegration of a localized perturbation into ion-acoustic solitons train in a plasma with hot electrons and cold ions. A Gaussian initial perturbation is used to model the localized perturbation. For this purpose, first, we reduce fluid system of equations to a Korteweg de-Vries equation by the following well-known assumptions. (i) On the ion-acoustic evolution time-scale, the electron velocity distribution function (EVDF) is assumed to be stationary. (ii) The calculation is restricted to small amplitude cases. Next, in order to generalize the model to finite amplitudes cases, the evolution of EVDF is included. To this end, a hybrid code is designed to simulate the case, in which electrons dynamics is governed by Vlasov equation, while cold ions dynamics is, like before, studied by the fluid equations. A comparison between the two models shows that although the fluid model is capable of demonstrating the general features of the process, to have a better insight into the relevant physics resulting from the evolution of EVDF, the use of kinetic treatment is of great importance.
NASA Astrophysics Data System (ADS)
Michael, Manesh; Willington, Neethu T.; Jayakumar, Neethu; Sebastian, Sijo; Sreekala, G.; Venugopal, Chandu
2016-12-01
We investigate the existence of ion-acoustic shock waves in a five component cometary plasma consisting of positively and negatively charged oxygen ions, kappa described hydrogen ions, hot solar electrons, and slightly colder cometary electrons. The KdVB equation has been derived for the system, and its solution plotted for different kappa values, oxygen ion densities, as well as the temperature ratios for the ions. It is found that the amplitude of the shock wave decreases with increasing kappa values. The strength of the shock profile decreases with increasing temperatures of the positively charged oxygen ions and densities of negatively charged oxygen ions.
Slunyaev, A V; Pelinovsky, E N
2016-11-18
The role of multiple soliton and breather interactions in the formation of very high waves is disclosed within the framework of the integrable modified Korteweg-de Vries (MKdV) equation. Optimal conditions for the focusing of many solitons are formulated explicitly. Namely, trains of ordered solitons with alternate polarities evolve to huge strongly localized transient waves. The focused wave amplitude is exactly the sum of the focusing soliton heights; the maximum wave inherits the polarity of the fastest soliton in the train. The focusing of several solitary waves or/and breathers may naturally occur in a soliton gas and will lead to rogue-wave-type dynamics; hence, it represents a new nonlinear mechanism of rogue wave generation. The discovered scenario depends crucially on the soliton polarities (phases), and is not taken into account by existing kinetic theories. The performance of the soliton mechanism of rogue wave generation is shown for the example of the focusing MKdV equation, when solitons possess "frozen" phases (certain polarities), though the approach is efficient in some other integrable systems which admit soliton and breather solutions.
NASA Astrophysics Data System (ADS)
Slunyaev, A. V.; Pelinovsky, E. N.
2016-11-01
The role of multiple soliton and breather interactions in the formation of very high waves is disclosed within the framework of the integrable modified Korteweg-de Vries (MKdV) equation. Optimal conditions for the focusing of many solitons are formulated explicitly. Namely, trains of ordered solitons with alternate polarities evolve to huge strongly localized transient waves. The focused wave amplitude is exactly the sum of the focusing soliton heights; the maximum wave inherits the polarity of the fastest soliton in the train. The focusing of several solitary waves or/and breathers may naturally occur in a soliton gas and will lead to rogue-wave-type dynamics; hence, it represents a new nonlinear mechanism of rogue wave generation. The discovered scenario depends crucially on the soliton polarities (phases), and is not taken into account by existing kinetic theories. The performance of the soliton mechanism of rogue wave generation is shown for the example of the focusing MKdV equation, when solitons possess "frozen" phases (certain polarities), though the approach is efficient in some other integrable systems which admit soliton and breather solutions.
NASA Astrophysics Data System (ADS)
Kumar, Sandeep; Tiwari, Sanat Kumar; Das, Amita
2017-03-01
The excitation and evolution of Korteweg-de Vries (KdV) solitons in a dusty plasma medium are studied using Molecular Dynamics (MD) simulations. The dusty plasma medium is modelled as a collection of dust particles interacting through Yukawa potential, which takes into account dust charge screening due to the lighter electron and ion species. The collective response of such screened dust particles to an applied electric field impulse is studied here. An excitation of a perturbed positive density pulse propagating in one direction along with a train of negative perturbed rarefactive density oscillations in the opposite direction is observed. These observations are in accordance with evolution governed by the KdV equation. Detailed studies of (a) amplitude vs. width variation of the observed pulse, (b) the emergence of intact separate pulses with an associated phase shift after collisional interaction amidst them, etc., conclusively qualify the positive pulses observed in the simulations as KdV solitons. It is also observed that by increasing the strength of the electric field impulse, multiple solitonic structures get excited. The excitations of the multiple solitons are similar to the experimental observations reported recently by Boruah et al. [Phys. Plasmas 23, 093704 (2016)] for dusty plasmas. The role of coupling parameter has also been investigated here, which shows that with increasing coupling parameter, the amplitude of the solitonic pulse increases whereas its width decreases.
NASA Astrophysics Data System (ADS)
Zhu, Quanyong; Fei, Jinxi; Ma, Zhengyi
2017-08-01
The nonlocal residual symmetry of a (2+1)-dimensional general Korteweg-de Vries (GKdV) system is derived by the truncated Painlevé analysis. The nonlocal residual symmetry is then localized to a Lie point symmetry by introducing auxiliary-dependent variables. By using Lie's first theorem, the finite transformation is obtained for the localized residual symmetry. Furthermore, multiple Bäcklund transformations are also obtained from the Lie point symmetry approach via the localization of the linear superpositions of multiple residual symmetries. As a result, various localized structures, such as dromion lattice, multiple-soliton solutions, and interaction solutions can be obtained through it; and these localized structures are illustrated by graphs.
NASA Astrophysics Data System (ADS)
Tian, Bo; Wei, Guang-Mei; Zhang, Chun-Yi; Shan, Wen-Rui; Gao, Yi-Tian
2006-07-01
The variable-coefficient Korteweg de Vries (KdV)-typed models, although often hard to be studied, are of current interest in describing various real situations. Under investigation hereby is a large class of the generalized variable-coefficient KdV models with external-force and perturbed/dissipative terms. Recent examples of this class include those in blood vessels and circulatory system, arterial dynamics, trapped Bose Einstein condensates related to matter waves and nonlinear atom optics, Bose gas of impenetrable bosons with longitudinal confinement, rods of compressible hyperelastic material and semiconductor heterostructures with positonic phenomena. In this Letter, based on symbolic computation, four transformations are proposed from this class either to the cylindrical or standard KdV equation when the respective constraint holds. The constraints have nothing to do with the external-force term. Under those transformations, such analytic solutions as those with the Airy, Hermit and Jacobian elliptic functions can be obtained, including the solitonic profiles. The roles for the perturbed and external-force terms to play are observed and discussed. Investigations on this class can be performed through the properties of solutions of cylindrical and standard KdV equations.
Duality relation among the Hamiltonian structures of a parametric coupled Korteweg-de Vries system
NASA Astrophysics Data System (ADS)
Restuccia, Alvaro; Sotomayor, Adrián
2016-01-01
We obtain the full Hamiltonian structure for a parametric coupled KdV system. The coupled system arises from four different real basic lagrangians. The associated Hamiltonian functionals and the corresponding Poisson structures follow from the geometry of a constrained phase space by using the Dirac approach for constrained systems. The overall algebraic structure for the system is given in terms of two pencils of Poisson structures with associated Hamiltonians depending on the parameter of the Poisson pencils. The algebraic construction we present admits the most general space of observables related to the coupled system. We then construct two master lagrangians for the coupled system whose field equations are the ɛ-parametric Gardner equations obtained from the coupled KdV system through a Gardner transformation. In the weak limit ɛ → 0 the lagrangians reduce to the ones of the coupled KdV system while, after a suitable redefinition of the fields, in the strong limit ɛ → ∞ we obtain the lagrangians of the coupled modified KdV system. The Hamiltonian structures of the coupled KdV system follow from the Hamiltonian structures of the master system by taking the two limits ɛ → 0 and ɛ → ∞.
Chen, Junchao; Xin, Xiangpeng; Chen, Yong
2014-05-15
The nonlocal symmetry is derived from the known Darboux transformation (DT) of the Hirota-Satsuma coupled Korteweg-de Vries (HS-cKdV) system, and infinitely many nonlocal symmetries are given by introducing the internal parameters. By extending the HS-cKdV system to an auxiliary system with five dependent variables, the prolongation is found to localize the so-called seed nonlocal symmetry related to the DT. By applying the general Lie point symmetry method to this enlarged system, we obtain two main results: a new type of finite symmetry transformation is derived, which is different from the initial DT and can generate new solutions from old ones; some novel exact interaction solutions among solitons and other complicated waves including periodic cnoidal waves and Painlevé waves are computed through similarity reductions. In addition, two kinds of new integrable models are proposed from the obtained nonlocal symmetry: the negative HS-cKdV hierarchy by introducing the internal parameters; the integrable models both in lower and higher dimensions by restricting the symmetry constraints.
Efficient Numerical Methods for Evolution Partial Differential Equations
1989-09-30
public lease; distribution mlim ed.-.... 13. ABSTRACT (Maxmum 200 woard Generalized Korteweg - de Vries equation (GKdV). This equation is written as...McKinney. On Optimal high-order in time approxiniations.for the Korteweg -de Vries equation ..To appear in Math. Comp.. 3. J.L. Bona, V.A. Dougalis...O.Karakashian and W. Mckinney, Conservative high-order schemes for the Generalized Korteweg -de Vries equation . In preparation. 4. G. D. Akrivis, V.A
NASA Astrophysics Data System (ADS)
Al-Akhaly, Galal A.; Dey, Bishwajyoti
2011-09-01
We show the existence of a type of excitation, which we term as “gap compactonlike,” within the gap of the linear spectrum of a system of coupled Kortweg-de Vries equations with linear and nonlinear dispersions. Since the solutions lie in the gap region of the spectra, they avoid resonance with the linear oscillatory wave and, therefore, do not decay into radiations. These types of solutions are important in energy localization and transport in polymers and biopolymers, optical systems, etc.
Averaging and renormalization for the Korteveg-deVries-Burgers equation.
Chorin, Alexandre J
2003-08-19
We consider traveling wave solutions of the Korteveg-deVries-Burgers equation and set up an analogy between the spatial averaging of these traveling waves and real-space renormalization for Hamiltonian systems. The result is an effective equation that reproduces means of the unaveraged, highly oscillatory, solution. The averaging enhances the apparent diffusion, creating an "eddy" (or renormalized) diffusion coefficient; the relation between the eddy diffusion coefficient and the original diffusion coefficient is found numerically to be one of incomplete similarity, setting up an instance of Barenblatt's renormalization group. The results suggest a relation between self-similar solutions of differential equations on one hand and renormalization groups and optimal prediction algorithms on the other. An analogy with hydrodynamics is pointed out.
Bridges, Thomas J.
2016-01-01
Multiphase wavetrains are multiperiodic travelling waves with a set of distinct wavenumbers and distinct frequencies. In conservative systems, such families are associated with the conservation of wave action or other conservation law. At generic points (where the Jacobian of the wave action flux is non-degenerate), modulation of the wavetrain leads to the dispersionless multiphase conservation of wave action. The main result of this paper is that modulation of the multiphase wavetrain, when the Jacobian of the wave action flux vector is singular, morphs the vector-valued conservation law into the scalar Korteweg–de Vries (KdV) equation. The coefficients in the emergent KdV equation have a geometrical interpretation in terms of projection of the vector components of the conservation law. The theory herein is restricted to two phases to simplify presentation, with extensions to any finite dimension discussed in the concluding remarks. Two applications of the theory are presented: a coupled nonlinear Schrödinger equation and two-layer shallow-water hydrodynamics with a free surface. Both have two-phase solutions where criticality and the properties of the emergent KdV equation can be determined analytically. PMID:28119546
Complex and singular solutions of KdV and MKdV equations
NASA Technical Reports Server (NTRS)
Buti, B.; Rao, N. N.; Khadkikar, S. B.
1986-01-01
The Korteweg-de Vries (KdV) and the modified Korteweg-de Vries (MKdV) equations are shown to have, besides the regular real solutions, exact regular complex as well as singular solutions. The singular solution for the KdV is real but for the MKdV it is pure imaginary. Implications of the complex solutions are discussed.
A Comparison of Solutions of Two Model Equations for Long Waves.
1983-02-01
focused on solutions of (A) and (B) that correspond to the initial condition that u(x,0) is a given function. Equation (A) is the Korteweg -de Vries...waves on the surface of water, two models have received particular attention. One is the equation of Korteweg and de Vries (1895) ( equation (A) or the...channel is the equation proposed by Korteweg and de Vries (1895), 3 1:’il nt * nx + 7 nnx + F n + . l= O. (la) In this equation n - n(x,t) represents
Mitlin, Vlad
2005-10-15
A new transformation termed the mu-derivative is introduced. Applying it to the Cahn-Hilliard equation yields dynamical exact solutions. It is shown that the mu-transformed Cahn-Hilliard equation can be presented in a separable form. This transformation also yields dynamical exact solutions and separable forms for other nonlinear models such as the modified Korteveg-de Vries and the Burgers equations. The general structure of a nonlinear partial differential equation that becomes separable upon applying the mu-derivative is described.
NASA Astrophysics Data System (ADS)
Giavedoni, Pietro
2017-03-01
We address the problem of long-time asymptotics for the solutions of the Korteweg–de Vries equation under low regularity assumptions. We consider decaying initial data admitting only a finite number of moments. For the so-called ‘soliton region’, an improved asymptotic estimate is provided, in comparison with the one in Grunert and Teschl (2009 Math. Phys. Anal. Geom. 12 287–324). Our analysis is based on the dbar steepest descent method proposed by Miller and McLaughlin. Dedicated to Dora, Paolo and Sanja, with deep gratitude for their love and support.
Adhikary, N. C.; Deka, M. K.; Dev, A. N.; Sarmah, J.
2014-08-15
In this report, the investigation of the properties of dust acoustic (DA) solitary wave propagation in an adiabatic dusty plasma including the effect of the non-thermal ions and trapped electrons is presented. The reductive perturbation method has been employed to derive the modified Korteweg–de Vries (mK-dV) equation for dust acoustic solitary waves in a homogeneous, unmagnetized, and collisionless plasma whose constituents are electrons, singly charged positive ions, singly charged negative ions, and massive charged dust particles. The stationary analytical solution of the mK-dV equation is numerically analyzed and where the effect of various dusty plasma constituents DA solitary wave propagation is taken into account. It is observed that both the ions in dusty plasma play as a key role for the formation of both rarefactive as well as the compressive DA solitary waves and also the ion concentration controls the transformation of negative to positive potentials of the waves.
Deriving average soliton equations with a perturbative method
Ballantyne, G.J.; Gough, P.T.; Taylor, D.P. )
1995-01-01
The method of multiple scales is applied to periodically amplified, lossy media described by either the nonlinear Schroedinger (NLS) equation or the Korteweg--de Vries (KdV) equation. An existing result for the NLS equation, derived in the context of nonlinear optical communications, is confirmed. The method is then applied to the KdV equation and the result is confirmed numerically.
NASA Astrophysics Data System (ADS)
Tian, Shou-Fu
2017-09-01
In this paper, we implement the Fokas method in order to study initial-boundary value problems of the coupled modified Korteweg–de Vries equation formulated on the half-line, with Lax pairs involving 3× 3 matrices. This equation can be considered as a generalization of the modified KdV equation. We show that the solution \\{ p(x, t), q(x, t)\\} can be written in terms of the solution of a 3× 3 Riemann–Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the matrix-value spectral functions s(k) and S(k) , which are respectively determined by the initial values and boundary values at x=0 . Finally, the associated Dirichlet to Neumann map of the equation is analyzed in detail. Some of these boundary values are unknown; however, using the fact that these specific functions satisfy a certain global relation, the unknown boundary values can be expressed in terms of the given initial and boundary data.
Optimal error estimates for high order Runge-Kutta methods applied to evolutionary equations
McKinney, W.R.
1989-01-01
Fully discrete approximations to 1-periodic solutions of the Generalized Korteweg de-Vries and the Cahn-Hilliard equations are analyzed. These approximations are generated by an Implicit Runge-Kutta method for the temporal discretization and a Galerkin Finite Element method for the spatial discretization. Furthermore, these approximations may be of arbitrarily high order. In particular, it is shown that the well-known order reduction phenomenon afflicting Implicit Runge Kutta methods does not occur. Numerical results supporting these optimal error estimates for the Korteweg-de Vries equation and indicating the existence of a slow motion manifold for the Cahn-Hilliard equation are also provided.
Differential geometry techniques for sets of nonlinear partial differential equations
NASA Technical Reports Server (NTRS)
Estabrook, Frank B.
1990-01-01
An attempt is made to show that the Cartan theory of partial differential equations can be a useful technique for applied mathematics. Techniques for finding consistent subfamilies of solutions that are generically rich and well-posed and for introducing potentials or other usefully consistent auxiliary fields are introduced. An extended sample calculation involving the Korteweg-de Vries equation is given.
Differential geometry techniques for sets of nonlinear partial differential equations
NASA Technical Reports Server (NTRS)
Estabrook, Frank B.
1990-01-01
An attempt is made to show that the Cartan theory of partial differential equations can be a useful technique for applied mathematics. Techniques for finding consistent subfamilies of solutions that are generically rich and well-posed and for introducing potentials or other usefully consistent auxiliary fields are introduced. An extended sample calculation involving the Korteweg-de Vries equation is given.
Bilinear approach to the supersymmetric Gardner equation
NASA Astrophysics Data System (ADS)
Babalic, C. N.; Carstea, A. S.
2016-08-01
We study a supersymmetric version of the Gardner equation (both focusing and defocusing) using the superbilinear formalism. This equation is new and cannot be obtained from the supersymmetric modified Korteweg-de Vries equation with a nonzero boundary condition. We construct supersymmetric solitons and then by passing to the long-wave limit in the focusing case obtain rational nonsingular solutions. We also discuss the supersymmetric version of the defocusing equation and the dynamics of its solutions.
Systems of Nonlinear Hyperbolic Partial Differential Equations
1997-12-01
McKinney) Travelling wave solutions of the modified Korteweg - deVries -Burgers Equation . J. Differential Equations , 116 (1995), 448-467. 4. (with D.G...SUBTITLE Systems of Nonlinear Hyperbolic Partial Differential Equations 6. AUTHOR’S) Michael Shearer PERFORMING ORGANIZATION NAMES(S) AND...DISTRIBUTION CODE 13. ABSTRACT (Maximum 200 words) This project concerns properties of wave propagation in partial differential equations that are nonlinear
A N = 2 extension of the Hirota bilinear formalism and the supersymmetric KdV equation
NASA Astrophysics Data System (ADS)
Delisle, Laurent
2017-01-01
We present a bilinear Hirota representation of the N = 2 supersymmetric extension of the Korteweg-de Vries equation. This representation is deduced using binary Bell polynomials, hierarchies, and fermionic limits. We, also, propose a new approach for the generalisation of the Hirota bilinear formalism in the N = 2 supersymmetric context.
On invariant analysis of some time fractional nonlinear systems of partial differential equations. I
NASA Astrophysics Data System (ADS)
Singla, Komal; Gupta, R. K.
2016-10-01
An investigation of Lie point symmetries for systems of time fractional partial differential equations including Ito system, coupled Burgers equations, coupled Korteweg de Vries equations, Hirota-Satsuma coupled KdV equations, and coupled nonlinear Hirota equations has been done. Using the obtained symmetries, each one of the systems is reduced to the nonlinear system of fractional ordinary differential equations involving Erdélyi-Kober fractional differential operator depending on a parameter α.
Das, Jayasree; Bandyopadhyay, Anup; Das, K. P.
2007-09-15
The purpose of this paper is to present the recent work of Das et al. [J. Plasma Phys. 72, 587 (2006)] on the existence and stability of the alternative solitary wave solution of fixed width of the combined MKdV-KdV-ZK (Modified Korteweg-de Vries-Korteweg-de Vries-Zakharov-Kuznetsov) equation for the ion-acoustic wave in a magnetized nonthermal plasma consisting of warm adiabatic ions in a more generalized form. Here we derive the alternative solitary wave solution of variable width instead of fixed width of the combined MKdV-KdV-ZK equation along with the condition for its existence and find that this solution assumes the sech profile of the MKdV-ZK (Modified Korteweg-de Vries-Zakharov-Kuznetsov) equation, when the coefficient of the nonlinear term of the KdV-ZK (Korteweg-de Vries-Zakharov-Kuznetsov) equation tends to zero. The three-dimensional stability analysis of the alternative solitary wave solution of variable width of the combined MKdV-KdV-ZK equation shows that the instability condition and the first order growth rate of instability are exactly the same as those of the solitary wave solution (the sech profile) of the MKdV-ZK equation, when the coefficient of the nonlinear term of the KdV-ZK equation tends to zero.
Painleve Chains for the Study of Integrable Higher Order Differential Equations.
1986-12-18
evolution equations , 1,2,3,4, 5 has become of special interest to theoretical physicists. Such equations possess a special type of elementary solution taking...diverse areas of physics including fluid dynamics, ferromagnetism, quantum optics, and crystal dislocations. Solution of important evolution equations ...and the most important evolution equations including the Burgers, Korteweg-de Vries ( KdV ), modified KdV , and Boussinesq equations . The present paper
Wang, Lei; Gao, Yi-Tian; Qi, Feng-Hua
2012-08-15
Under investigation in this paper is a variable-coefficient modified Kortweg-de Vries (vc-mKdV) model describing certain situations from the fluid mechanics, ocean dynamics and plasma physics. N-fold Darboux transformation (DT) of a variable-coefficient Ablowitz-Kaup-Newell-Segur spectral problem is constructed via a gauge transformation. Multi-solitonic solutions in terms of the double Wronskian for the vc-mKdV model are derived by the reduction of the N-fold DT. Three types of the solitonic interactions are discussed through figures: (1) Overtaking collision; (2) Head-on collision; (3) Parallel solitons. Nonlinear, dispersive and dissipative terms have the effects on the velocities of the solitonic waves while the amplitudes of the waves depend on the perturbation term. - Highlights: Black-Right-Pointing-Pointer N-fold DT is firstly applied to a vc-AKNS spectral problem. Black-Right-Pointing-Pointer Seeking a double Wronskian solution is changed into solving two systems. Black-Right-Pointing-Pointer Effects of the variable coefficients on the multi-solitonic waves are discussed in detail. Black-Right-Pointing-Pointer This work solves the problem from Yi Zhang [Ann. Phys. 323 (2008) 3059].
Integrability of the Kruskal--Zabusky Discrete Equation by Multiscale Expansion
Levi, Decio; Scimiterna, Christian
2010-03-08
In 1965 Kruskal and Zabusky in a very famous article in Physical Review Letters introduced the notion of 'soliton' to describe the interaction of solitary waves solutions of the Korteweg de Vries equation (KdV). To do so they introduced a discrete approximation to the KdV, the Kruskal-Zabusky equation (KZ). Here we analyze the KZ equation using the multiscale expansion and show that the equation is only A{sub 2} integrable.
Linear superposition in nonlinear equations.
Khare, Avinash; Sukhatme, Uday
2002-06-17
Several nonlinear systems such as the Korteweg-de Vries (KdV) and modified KdV equations and lambda phi(4) theory possess periodic traveling wave solutions involving Jacobi elliptic functions. We show that suitable linear combinations of these known periodic solutions yield many additional solutions with different periods and velocities. This linear superposition procedure works by virtue of some remarkable new identities involving elliptic functions.
Genetics Home Reference: Koolen-de Vries syndrome
... including the KANSL1 gene, from one copy of chromosome 17 . This type of genetic abnormality is called a microdeletion. A small number of individuals with Koolen-de Vries syndrome do not have a chromosome 17 microdeletion but instead have a mutation within ...
1981-01-08
as it propagates over a small interval, and then to correct for absorption. Another nonlinear wave equation of great interest is the Korteweg - DeVries ...acoustics are described by the second-order-nonlinear wave equation , which is derived in this thesis and solved by numerical means. the validity of the...no approximations are made in the second-order-nonlinear acoustic wave equation as it is solved . This represents an advance on the prior art, in which
On the nature of the de Vries smectic-A liquid crystal phase
NASA Astrophysics Data System (ADS)
Maclennan, Joseph; Zhu, Chenhui; Shen, Yongqiang; Shao, Renfan; Glaser, Matthew; Clark, Noel; Walba, David; Korblova, Eva; Moran, Mark; Yang, Hong; Wang, Lixing; Pindak, Ron; Lemieux, Robert
2010-03-01
Recently a new model for the de Vries SmA (``sugar cone' model) has been proposed.footnotetextS. T. Lagerwall, P. Rudquist, F. Giesselmann, ``The orientational order in so-called de Vries materials'', Mol. Cryst. Liq. Cryst. 510, 1282 (2009). We present results from three different x-ray diffraction experiments designed to explore the nature of the de Vries smectic-A phase: (1) comparison of the first- and second-order smectic Bragg reflections across the smectic-A-smectic-C phase transition; (2) resonant scattering to probe anticlinic correlations in the de Vries smectic-A phase, and (3) measurements of the variation of layer spacing in the de Vries SmA and SmC phases with applied electric field. These experiments may help distinguish the well-known ``hollow cone'' modelfootnotetextA. de Vries, Mol. Cryst. Liq. Cryst. 41, 27 (1977). from the ``sugar cone'' model.
Fast neural solution of a nonlinear wave equation
NASA Technical Reports Server (NTRS)
Toomarian, Nikzad; Barhen, Jacob
1992-01-01
A neural algorithm for rapidly simulating a certain class of nonlinear wave phenomena using analog VLSI neural hardware is presented and applied to the Korteweg-de Vries partial differential equation. The corresponding neural architecture is obtained from a pseudospectral representation of the spatial dependence, along with a leap-frog scheme for the temporal evolution. Numerical simulations demonstrated the robustness of the proposed approach.
Fast neural solution of a nonlinear wave equation
NASA Technical Reports Server (NTRS)
Toomarian, Nikzad; Barhen, Jacob
1992-01-01
A neural algorithm for rapidly simulating a certain class of nonlinear wave phenomena using analog VLSI neural hardware is presented and applied to the Korteweg-de Vries partial differential equation. The corresponding neural architecture is obtained from a pseudospectral representation of the spatial dependence, along with a leap-frog scheme for the temporal evolution. Numerical simulations demonstrated the robustness of the proposed approach.
Two-component coupled KdV equations and its connection with the generalized Harry Dym equations
Popowicz, Ziemowit
2014-01-15
It is shown that three different Lax operators in the Dym hierarchy produce three generalized coupled Harry Dym equations. These equations transform, via the reciprocal link, to the coupled two-component Korteweg de Vries (KdV) system. The first equation gives us known integrable two-component KdV system, while the second reduces to the known symmetrical two-component KdV equation. The last one reduces to the Drienfeld-Sokolov equation. This approach gives us new Lax representation for these equations.
Two-component coupled KdV equations and its connection with the generalized Harry Dym equations
NASA Astrophysics Data System (ADS)
Popowicz, Ziemowit
2014-01-01
It is shown that three different Lax operators in the Dym hierarchy produce three generalized coupled Harry Dym equations. These equations transform, via the reciprocal link, to the coupled two-component Korteweg de Vries (KdV) system. The first equation gives us known integrable two-component KdV system, while the second reduces to the known symmetrical two-component KdV equation. The last one reduces to the Drienfeld-Sokolov equation. This approach gives us new Lax representation for these equations.
Prolongation structures of nonlinear evolution equations
NASA Technical Reports Server (NTRS)
Wahlquist, H. D.; Estabrook, F. B.
1975-01-01
A technique is developed for systematically deriving a 'prolongation structure' - a set of interrelated potentials and pseudopotentials - for nonlinear partial differential equations in two independent variables. When this is applied to the Korteweg-de Vries equation, a new infinite set of conserved quantities is obtained. Known solution techniques are shown to result from the discovery of such a structure: related partial differential equations for the potential functions, linear 'inverse scattering' equations for auxiliary functions, Backlund transformations. Generalizations of these techniques will result from the use of irreducible matrix representations of the prolongation structure.
Formation of wave packets in the Ostrovsky equation for both normal and anomalous dispersion.
Grimshaw, Roger; Stepanyants, Yury; Alias, Azwani
2016-01-01
It is well known that the Ostrovsky equation with normal dispersion does not support steady solitary waves. An initial Korteweg-de Vries solitary wave decays adiabatically through the radiation of long waves and is eventually replaced by an envelope solitary wave whose carrier wave and envelope move with different velocities (phase and group velocities correspondingly). Here, we examine the same initial condition for the Ostrovsky equation with anomalous dispersion, when the wave frequency increases with wavenumber in the limit of very short waves. The essential difference is that now there exists a steady solitary wave solution (Ostrovsky soliton), which in the small-amplitude limit can be described asymptotically through the solitary wave solution of a nonlinear Schrödinger equation, based at that wavenumber where the phase and group velocities coincide. Long-time numerical simulations show that the emergence of this steady envelope solitary wave is a very robust feature. The initial Korteweg-de Vries solitary wave transforms rapidly to this envelope solitary wave in a seemingly non-adiabatic manner. The amplitude of the Ostrovsky soliton strongly correlates with the initial Korteweg-de Vries solitary wave.
Five-wave classical scattering matrix and integrable equations
NASA Astrophysics Data System (ADS)
Zakharov, V. E.; Odesskii, A. V.; Cisternino, M.; Onorato, M.
2014-07-01
We study the five-wave classical scattering matrix for nonlinear and dispersive Hamiltonian equations with a nonlinearity of the type u∂u/∂x. Our aim is to find the most general nontrivial form of the dispersion relation ω(k) for which the five-wave interaction scattering matrix is identically zero on the resonance manifold. As could be expected, the matrix in one dimension is zero for the Korteweg-de Vries equation, the Benjamin-Ono equation, and the intermediate long-wave equation. In two dimensions, we find a new equation that satisfies our requirement.
Islam, Md. Shafiqul; Khan, Kamruzzaman; Akbar, M. Ali; Mastroberardino, Antonio
2014-01-01
The purpose of this article is to present an analytical method, namely the improved F-expansion method combined with the Riccati equation, for finding exact solutions of nonlinear evolution equations. The present method is capable of calculating all branches of solutions simultaneously, even if multiple solutions are very close and thus difficult to distinguish with numerical techniques. To verify the computational efficiency, we consider the modified Benjamin–Bona–Mahony equation and the modified Korteweg-de Vries equation. Our results reveal that the method is a very effective and straightforward way of formulating the exact travelling wave solutions of nonlinear wave equations arising in mathematical physics and engineering. PMID:26064530
An asymptotic-preserving method for a relaxation of the Navier-Stokes-Korteweg equations
NASA Astrophysics Data System (ADS)
Chertock, Alina; Degond, Pierre; Neusser, Jochen
2017-04-01
The Navier-Stokes-Korteweg (NSK) equations are a classical diffuse-interface model for compressible two-phase flows. As direct numerical simulations based on the NSK system are quite expensive and in some cases even impossible, we consider a relaxation of the NSK system, for which robust numerical methods can be designed. However, time steps for explicit numerical schemes depend on the relaxation parameter and therefore numerical simulations in the relaxation limit are very inefficient. To overcome this restriction, we propose an implicit-explicit asymptotic-preserving finite volume method. We prove that the new scheme provides a consistent discretization of the NSK system in the relaxation limit and demonstrate that it is capable of accurately and efficiently computing numerical solutions of problems with realistic density ratios and small interfacial widths.
Solitons induced by boundary conditions from the Boussinesq equation
NASA Technical Reports Server (NTRS)
Chou, Ru Ling; Chu, C. K.
1990-01-01
The behavior of solitons induced by boundary excitation is investigated at various time-dependent conditions and different unperturbed water depths, using the Korteweg-de Vries (KdV) equation. Then, solitons induced from Boussinesq equations under similar conditions were studied, making it possible to remove the restriction in the KdV equation and to treat soliton head-on collisions (as well as overtaking collisions) and reflections. It is found that the results obtained from the KdV and the Boussinesq equations are in good agreement.
An integrable shallow water equation with linear and nonlinear dispersion.
Dullin, H R; Gottwald, G A; Holm, D D
2001-11-05
We use asymptotic analysis and a near-identity normal form transformation from water wave theory to derive a 1+1 unidirectional nonlinear wave equation that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation. This equation is one order more accurate in asymptotic approximation beyond KdV, yet it still preserves complete integrability via the inverse scattering transform method. Its traveling wave solutions contain both the KdV solitons and the CH peakons as limiting cases.
Stability for line solitary waves of Zakharov-Kuznetsov equation
NASA Astrophysics Data System (ADS)
Yamazaki, Yohei
2017-04-01
In this paper, we consider the stability for line solitary waves of the two dimensional Zakharov-Kuznetsov equation on R ×TL which is one of a high dimensional generalization of Korteweg-de Vries equation, where TL is the torus with the 2 πL period. The orbital and asymptotic stability of the one soliton of Korteweg-de Vries equation on the energy space was proved by Benjamin [2], Pego and Weinstein [41] and Martel and Merle [30]. We regard the one soliton of Korteweg-de Vries equation as a line solitary wave of Zakharov-Kuznetsov equation on R ×TL. We prove the stability and the transverse instability of the line solitary waves of Zakharov-Kuznetsov equation by applying the method of Evans' function and the argument of Rousset and Tzvetkov [44]. Moreover, we prove the asymptotic stability for orbitally stable line solitary waves of Zakharov-Kuznetsov equation by using the argument of Martel and Merle [30-32] and a Liouville type theorem. If L is the critical period with respect to a line solitary wave, the line solitary wave is orbitally stable. However, since this line solitary wave is a bifurcation point of the stationary equation, the linearized operator of the stationary equation is degenerate. Because of the degeneracy of the linearized operator, we can not show the Liouville type theorem for the line solitary wave by using the usual virial type estimate. To show the Liouville type theorem for the line solitary wave, we modify a virial type estimate.
An h-adaptive local discontinuous Galerkin method for the Navier-Stokes-Korteweg equations
NASA Astrophysics Data System (ADS)
Tian, Lulu; Xu, Yan; Kuerten, J. G. M.; van der Vegt, J. J. W.
2016-08-01
In this article, we develop a mesh adaptation algorithm for a local discontinuous Galerkin (LDG) discretization of the (non)-isothermal Navier-Stokes-Korteweg (NSK) equations modeling liquid-vapor flows with phase change. This work is a continuation of our previous research, where we proposed LDG discretizations for the (non)-isothermal NSK equations with a time-implicit Runge-Kutta method. To save computing time and to capture the thin interfaces more accurately, we extend the LDG discretization with a mesh adaptation method. Given the current adapted mesh, a criterion for selecting candidate elements for refinement and coarsening is adopted based on the locally largest value of the density gradient. A strategy to refine and coarsen the candidate elements is then provided. We emphasize that the adaptive LDG discretization is relatively simple and does not require additional stabilization. The use of a locally refined mesh in combination with an implicit Runge-Kutta time method is, however, non-trivial, but results in an efficient time integration method for the NSK equations. Computations, including cases with solid wall boundaries, are provided to demonstrate the accuracy, efficiency and capabilities of the adaptive LDG discretizations.
A Method of Solution for Painleve Equations: Painleve IV, V,
1987-02-01
Korteweg - deVries (KdV) equation lead to PI and PII [9); PHI and special cases of PIll and PIV can be obtained from the exact similarity reduction of the...value problem; solving such an initial value problem is essentially equivalent to solving an inverse problem for a certain isomonodromic linear equation ...there is a unified approach to solving certain initial value problems for equations in 1, 1+1 (one spatial and one temporal) and 2+1 dimensions. Using
1987-08-01
solution of the Korteweg-de Vries equation ( KdV ), working our way up to the derivation of the multi-soliton solution of the sine-Gordon equation (sG...SOLITARY WAVE SOLUTIONS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS j DiS~~Uj~l. _’UDistribution/Willy Hereman AvaiiLi -itY Codes Technical Summary Report...Key Words: soliton theory, solitary waves, coupled KdV , evolution equations , direct methods, Harry Dym, sine-Gordon Mathematics Department, University
Modulational instability in nonlinear nonlocal equations of regularized long wave type
NASA Astrophysics Data System (ADS)
Hur, Vera Mikyoung; Pandey, Ashish Kumar
2016-06-01
We study the stability and instability of periodic traveling waves in the vicinity of the origin in the spectral plane, for equations of Benjamin-Bona-Mahony (BBM) and regularized Boussinesq types permitting nonlocal dispersion. We extend recent results for equations of Korteweg-de Vries type and derive modulational instability indices as functions of the wave number of the underlying wave. We show that a sufficiently small, periodic traveling wave of the BBM equation is spectrally unstable to long wavelength perturbations if the wave number is greater than a critical value and a sufficiently small, periodic traveling wave of the regularized Boussinesq equation is stable to square integrable perturbations.
A Riemann-Hilbert Approach for the Novikov Equation
NASA Astrophysics Data System (ADS)
Boutet de Monvel, Anne; Shepelsky, Dmitry; Zielinski, Lech
2016-09-01
We develop the inverse scattering transform method for the Novikov equation u_t-u_{txx}+4u^2u_x=3u u_xu_{xx}+u^2u_{xxx} considered on the line xin(-∞,∞) in the case of non-zero constant background. The approach is based on the analysis of an associated Riemann-Hilbert (RH) problem, which in this case is a 3× 3 matrix problem. The structure of this RH problem shares many common features with the case of the Degasperis-Procesi (DP) equation having quadratic nonlinear terms (see [Boutet de Monvel A., Shepelsky D., Nonlinearity 26 (2013), 2081-2107, arXiv:1107.5995]) and thus the Novikov equation can be viewed as a ''modified DP equation'', in analogy with the relationship between the Korteweg-de Vries (KdV) equation and the modified Korteweg-de Vries (mKdV) equation. We present parametric formulas giving the solution of the Cauchy problem for the Novikov equation in terms of the solution of the RH problem and discuss the possibilities to use the developed formalism for further studying of the Novikov equation.
NASA Technical Reports Server (NTRS)
Ying, S. J.; Liu, V. C.
1978-01-01
The numerical scheme for the computation of a shock discontinuity developed by MacCormack has been extended to solve a number of differential equations, including cases explicitly containing higher-order derivatives: (1) Korteweg-de Vries equation with a term of third-order derivative, (2) a system of nonlinear equations governing nonsteady one-dimensional plasma flow in cylindrical coordinate, (3) equations of solar wind. Comparisons with previous results are made, if available, to illustrate the advantages of the present method. The question of convergence of the numerical calculation is discussed.
Korteveg-de Vries solitons in a cold quark-gluon plasma
NASA Astrophysics Data System (ADS)
Fogaça, D. A.; Navarra, F. S.; Ferreira Filho, L. G.
2011-09-01
The relativistic heavy ion program developed at RHIC and now at LHC motivated a deeper study of the properties of the quark-gluon plasma (QGP) and, in particular, the study of perturbations in this kind of plasma. We are interested on the time evolution of perturbations in the baryon and energy densities. If a localized pulse in baryon density could propagate throughout the QGP for long distances preserving its shape and without loosing localization, this could have interesting consequences for relativistic heavy ion physics and for astrophysics. A mathematical way to prove that this can happen is to derive (under certain conditions) from the hydrodynamical equations of the QGP a Korteveg-de Vries (KdV) equation. The solution of this equation describes the propagation of a KdV soliton. The derivation of the KdV equation depends crucially on the equation of state (EOS) of the QGP. The use of the simple MIT bag model EOS does not lead to KdV solitons. Recently we have developed an EOS for the QGP which includes both perturbative and nonperturbative corrections to the MIT one and is still simple enough to allow for analytical manipulations. With this EOS we were able to derive a KdV equation for the cold QGP.
Domains, defects, and de Vries: Electrooptics of smectic liquid crystals
NASA Astrophysics Data System (ADS)
Jones, Christopher D.
Liquid crystal (LC) materials are easily manipulated with the introduction of fields. Surface alignment of LC materials is commonly achieved via a rubbed polymer. Electric fields are then applied across the LC in order to reorient the individual molecules. These two controlling fields are the fundamental basis for the entirety of the liquid crystal display (LCD) industry, which in the 1970s was limited to calculators and digital watches but now LCDs are present by the dozen in the average home! Because these manipulations are so simple, and because the applications are so obvious, it has been useful to exploit the display cell geometry for the study of LCs. Novel compounds are being synthesized by chemistry groups at a high rate, and characterization of new materials must keep up. Therefore a primary technique is to probe the electrooptics of a material in a display cell. However, this geometry has its own impact on the behavior of a material: orientation and pinning at the surfaces tend to dominate the rest of the cell volume. With this information in mind, three interesting results of the display cell geometry and the resultant electrooptic measurements will be shown. First, the nucleation of twisted domains in achiral materials, made possible by the high energies required to overcome the orientation of the surface layers as compared to the bulk will be discussed. Second, the foundations of a large scale texture, made possible by surface pinning, expressing the stress of a material that shows large layer expansion on cooling through the smectic A phase will be solved. Finally, a model for the frequency dependence of the unique electrooptical behavior of the de Vries-type of smectics will be provided.
Exact solutions to the KDV-Burgers equation with forcing term using Tanh-Coth method
NASA Astrophysics Data System (ADS)
Chukkol, Yusuf Buba; Mohamad, Mohd Nor; Muminov, Mukhiddin I.
2017-08-01
In this paper, tanh-coth method was applied to derive the exact travelling wave solutions to the Korteweg-de-Vries and Burgers equation with forcing term(fKDVB). Solutions that are linear combination of solitary and shock wave solutions, and periodic wave solutions are obtained, by reducing the equation to the homogeneous type using a wave transformation. The method with the help of symbolic computation tool box provides a systematic way of solving many physical models involving nonlinear partial differential equations in mathematical physics.
NASA Astrophysics Data System (ADS)
Matsuno, Yoshimasa
2004-12-01
We present the new representations of the multiperiodic and multisoliton solutions of the Benjamin-Ono and nonlocal nonlinear Schrödinger equations. The key idea in the analysis is to explore the structure of the determinantal expressions of the solutions. After providing a direct verification of the multiperiodic solution by means of an elementary theory of determinants, we show that the solution admits a representation in terms of solutions for a system of nonlinear algebraic equations. This representation is found to be an analog of the multiperiodic solution of the Korteweg-de Vries equation. We also discuss the long-wave limit of the results associated with the multiperiodic solutions.
Soliton solutions of the KdV equation with higher-order corrections
NASA Astrophysics Data System (ADS)
Wazwaz, Abdul-Majid
2010-10-01
In this work, the Korteweg-de Vries (KdV) equation with higher-order corrections is examined. We studied the KdV equation with first-order correction and that with second-order correction that include the terms of the fifth-order Lax, Sawada-Kotera and Caudrey-Dodd-Gibbon equations. The simplified form of the bilinear method was used to show the integrability of the first-order models and therefore to obtain multiple soliton solutions for each one. The obstacles to integrability of some of the models with second-order corrections are examined as well.
Complex PT-symmetric nonlinear Schrödinger equation and Burgers equation.
Yan, Zhenya
2013-04-28
The complex -symmetric nonlinear wave models have drawn much attention in recent years since the complex -symmetric extensions of the Korteweg-de Vries (KdV) equation were presented in 2007. In this review, we focus on the study of the complex -symmetric nonlinear Schrödinger equation and Burgers equation. First of all, we briefly introduce the basic property of complex symmetry. We then report on exact solutions of one- and two-dimensional nonlinear Schrödinger equations (known as the Gross-Pitaevskii equation in Bose-Einstein condensates) with several complex -symmetric potentials. Finally, some complex -symmetric extension principles are used to generate some complex -symmetric nonlinear wave equations starting from both -symmetric (e.g. the KdV equation) and non- -symmetric (e.g. the Burgers equation) nonlinear wave equations. In particular, we discuss exact solutions of some representative ones of the complex -symmetric Burgers equation in detail.
Complementary studies of de Vries type SmA ∗ phase
NASA Astrophysics Data System (ADS)
Mikułko, A.; Marzec, M.; Wróbel, S.; Przedmojski, J.; Douali, R.; Legrand, Ch.; Dąbrowski, R.; Haase, W.
2006-11-01
Two chiral liquid crystalline compounds have been investigated, namely 4-(1-methylheptyloxycarbonyl) phenyl-4'nonylbiphenyl-4-carboxylate (MHPNBC), 3-(2-fluor-octyloxy)-6-(4octyl-phenyl) pyridine (FOOPP) to study the SmA ∗-SmC ∗ transition. For both substances strong electroclinic effect is observed in the SmA ∗ phase what indicates that it is the so-called de Vries SmA ∗ phase. To study the paralectric-ferroelectric de Vries type transition electrooptic, dielectric as well as SAXS methods have been applied.
1981-04-01
Contract No. DAAG29- 10-(’-w4l and by an A.M.S. Postdoctoral Research Fellowship. I SIGNIFICANCE AND EXPLANATION The Korteweg - deVries equation (KdV for...decomposition of the solution resembling the use of Fourier transforms in solving constant coefficient equations . For the linearization about the zero...1.3) to eliminate (f 2)I(x,k), it is more convenient not to do so.) We prove the theorem by solving the equation - 4Q’ - 2Q’ + 4k 2 ’ for ’ and
Undular bore theory for the Gardner equation.
Kamchatnov, A M; Kuo, Y-H; Lin, T-C; Horng, T-L; Gou, S-C; Clift, R; El, G A; Grimshaw, R H J
2012-09-01
We develop modulation theory for undular bores (dispersive shock waves) in the framework of the Gardner, or extended Korteweg-de Vries (KdV), equation, which is a generic mathematical model for weakly nonlinear and weakly dispersive wave propagation, when effects of higher order nonlinearity become important. Using a reduced version of the finite-gap integration method we derive the Gardner-Whitham modulation system in a Riemann invariant form and show that it can be mapped onto the well-known modulation system for the Korteweg-de Vries equation. The transformation between the two counterpart modulation systems is, however, not invertible. As a result, the study of the resolution of an initial discontinuity for the Gardner equation reveals a rich phenomenology of solutions which, along with the KdV-type simple undular bores, include nonlinear trigonometric bores, solibores, rarefaction waves, and composite solutions representing various combinations of the above structures. We construct full parametric maps of such solutions for both signs of the cubic nonlinear term in the Gardner equation. Our classification is supported by numerical simulations.
Cylindrical electron acoustic solitons for modified time-fractional nonlinear equation
NASA Astrophysics Data System (ADS)
Abdelwahed, H. G.; El-Shewy, E. K.; Mahmoud, Abeer A.
2017-08-01
The modulation of cylindrical electron acoustic characteristics using a time fractal modified nonlinear equation has been investigated in nonisothermal plasmas. The time fractional cylindrical modified-Korteweg-de Vries equation has been obtained by Agrawal's analysis. A cylindrical localized soliton has been obtained via the Adomian decomposition method. The pressure term and cylindrical time fractional effects on the modulated wave properties have been investigated with comparative auroral observations. It is established that the presence of the fractional order factor not only significantly modifies the solitary characteristics but also varies the profile polarity.
NASA Astrophysics Data System (ADS)
Trogdon, Thomas; Deconinck, Bernard
2013-05-01
We derive a Riemann-Hilbert problem satisfied by the Baker-Akhiezer function for the finite-gap solutions of the Korteweg-de Vries (KdV) equation. As usual for Riemann-Hilbert problems associated with solutions of integrable equations, this formulation has the benefit that the space and time dependence appears in an explicit, linear and computable way. We make use of recent advances in the numerical solution of Riemann-Hilbert problems to produce an efficient and uniformly accurate numerical method for computing all periodic and quasi-periodic finite-genus solutions of the KdV equation.
NASA Astrophysics Data System (ADS)
Liu, Lei; Tian, Bo; Sun, Wen-Rong; Wang, Yu-Feng; Wang, Yun-Po
2016-01-01
The transition phenomenon of few-cycle-pulse optical solitons from a pure modified Korteweg-de Vries (mKdV) to a pure sine-Gordon regime can be described by the nonautonomous mKdV-sinh-Gordon equation with time-dependent coefficients. Based on the Bell polynomials, Hirota method and symbolic computation, bilinear forms and soliton solutions for this equation are obtained. Bäcklund transformations (BTs) in both the binary Bell polynomial and bilinear forms are obtained. By virtue of the BTs and Ablowitz-Kaup-Newell-Segur system, Lax pair and infinitely many conservation laws for this equation are derived as well.
Linking Literacy, Technology, and the Environment: An Interview with Joan Goble and Rene de Vries.
ERIC Educational Resources Information Center
Strangman, Nicole
2003-01-01
Interviews Joan Goble, a third-grade teacher in Indiana, and Rene de Vries, a sixth-grade teacher in The Netherlands. Explains that the two teachers created and managed three Internet projects discussing endangered species and the environment. Notes that through these projects, students can experience the double satisfaction of educating others…
Linking Literacy, Technology, and the Environment: An Interview with Joan Goble and Rene de Vries.
ERIC Educational Resources Information Center
Strangman, Nicole
2003-01-01
Interviews Joan Goble, a third-grade teacher in Indiana, and Rene de Vries, a sixth-grade teacher in The Netherlands. Explains that the two teachers created and managed three Internet projects discussing endangered species and the environment. Notes that through these projects, students can experience the double satisfaction of educating others…
The Linear KdV Equation with an Interface
NASA Astrophysics Data System (ADS)
Deconinck, Bernard; Sheils, Natalie E.; Smith, David A.
2016-10-01
The interface problem for the linear Korteweg-de Vries (KdV) equation in one-dimensional piecewise homogeneous domains is examined by constructing an explicit solution in each domain. The location of the interface is known and a number of compatibility conditions at the boundary are imposed. We provide an explicit characterization of sufficient interface conditions for the construction of a solution using Fokas's Unified Transform Method. The problem and the method considered here extend that of earlier papers to problems with more than two spatial derivatives.
Bäcklund Transformations and Non-Abelian Nonlinear Evolution Equations: a Novel Bäcklund Chart
NASA Astrophysics Data System (ADS)
Carillo, Sandra; Lo Schiavo, Mauro; Schiebold, Cornelia
2016-08-01
Classes of third order non-Abelian evolution equations linked to that of Korteweg-de Vries-type are investigated and their connections represented in a non-commutative Bäcklund chart, generalizing results in [Fuchssteiner B., Carillo S., Phys. A 154 (1989), 467-510]. The recursion operators are shown to be hereditary, thereby allowing the results to be extended to hierarchies. The present study is devoted to operator nonlinear evolution equations: general results are presented. The implied applications referring to finite-dimensional cases will be considered separately.
Numerical study of blow-up and stability of line solitons for the Novikov-Veselov equation
NASA Astrophysics Data System (ADS)
Kazeykina, Anna; Klein, Christian
2017-07-01
We study numerically the evolution of perturbed Korteweg-de Vries solitons and of well localized initial data by the Novikov-Veselov (NV) equation at different levels of the ‘energy’ parameter E. We show that as \\vert E\\vert \\to ∞ , NV behaves, as expected, similarly to its formal limit, the Kadomtsev-Petviashvili equation. However at intermediate regimes, i.e. when \\vert E \\vert is not very large, more varied scenarios are possible, in particular, blow-ups are observed. The mechanism of the blow-up is studied.
Ryu, Seong Ho; Shin, Tae Joo; Gong, Tao; Shen, Yongqiang; Korblova, Eva; Shao, Renfan; Walba, David M; Clark, Noel A; Yoon, Dong Ki
2014-03-01
We have identified a metastable liquid-crystal (LC) structure in the de Vries smectic-A* phase (de Vries Sm-A*) formed by silicon-containing molecules under certain boundary conditions. The phase transition with the metastable structure was observed in a LC droplet placed on a planar aligned substrate and LCs confined in the groove of a silicon microchannel. During the rapid cooling step, a batonnet structure was generated as an intermediate and metastable state prior to the transition that yielded the thermodynamically stable toric focal conic domains. This distinctive behavior was characterized using depolarized reflection light microscopy and grazing incidence x-ray diffraction techniques. We concluded that the silicon groups in the molecules that formed the de Vries phase induced the formation of layered clusters called cybotactic structures. This observation is relevant to an exploration of the physical properties of cybotactic de Vries phases and gives a hint as to their optoelectronic applications.
NASA Astrophysics Data System (ADS)
Ryu, Seong Ho; Shin, Tae Joo; Gong, Tao; Shen, Yongqiang; Korblova, Eva; Shao, Renfan; Walba, David M.; Clark, Noel A.; Yoon, Dong Ki
2014-03-01
We have identified a metastable liquid-crystal (LC) structure in the de Vries smectic-A* phase (de Vries Sm-A*) formed by silicon-containing molecules under certain boundary conditions. The phase transition with the metastable structure was observed in a LC droplet placed on a planar aligned substrate and LCs confined in the groove of a silicon microchannel. During the rapid cooling step, a batonnet structure was generated as an intermediate and metastable state prior to the transition that yielded the thermodynamically stable toric focal conic domains. This distinctive behavior was characterized using depolarized reflection light microscopy and grazing incidence x-ray diffraction techniques. We concluded that the silicon groups in the molecules that formed the de Vries phase induced the formation of layered clusters called cybotactic structures. This observation is relevant to an exploration of the physical properties of cybotactic de Vries phases and gives a hint as to their optoelectronic applications.
NASA Astrophysics Data System (ADS)
Liu, Jian-Guo; Du, Jian-Qiang; Zeng, Zhi-Fang; Ai, Guo-Ping
2016-10-01
The Korteweg-de Vries (KdV)-type models have been shown to describe many important physical situations such as fluid flows, plasma physics, and solid state physics. In this paper, a new (2 + 1)-dimensional KdV equation is discussed. Based on the Hirota's bilinear form and a generalized three-wave approach, we obtain new exact solutions for the new (2 + 1)-dimensional KdV equation. With the help of symbolic computation, the properties for some new solutions are presented with some figures.
Liu, Jian-Guo; Du, Jian-Qiang; Zeng, Zhi-Fang; Ai, Guo-Ping
2016-10-01
The Korteweg-de Vries (KdV)-type models have been shown to describe many important physical situations such as fluid flows, plasma physics, and solid state physics. In this paper, a new (2 + 1)-dimensional KdV equation is discussed. Based on the Hirota's bilinear form and a generalized three-wave approach, we obtain new exact solutions for the new (2 + 1)-dimensional KdV equation. With the help of symbolic computation, the properties for some new solutions are presented with some figures.
Absent bystanders and cognitive dissonance: a comment on Timmins & de Vries.
Paley, John
2015-04-01
Timmins & de Vries are more sympathetic to my editorial than other critics, but they take issue with the details. They doubt whether the bystander phenomenon applies to Mid Staffs nurses; they believe that cognitive dissonance is a better explanation of why nurses fail to behave compassionately; and they think that I am 'perhaps a bit rash' to conclude that 'teaching compassion may be fruitless'. In this comment, I discuss all three points. I suggest that the bystander phenomenon is irrelevant; that Timmins & de Vries give an incomplete account of cognitive dissonance; and that it isn't rash to propose that educating nurses 'for compassion' is a red herring. Additionally, I comment on the idea that I wish to mount a 'defence of healthcare staff'. Copyright © 2014 Elsevier Ltd. All rights reserved.
Singular Perturbation Methods for Nonlinear Dynamical Systems and Waves
1992-07-01
Korteweg -de Vries equation [10] 2. Structure of two-dimensional diffusive shock waves [1] In addition, preliminary work began on two problems: 1...oscillatory waves. 3. Korteweg -de Vries equation . In [41 these ideas were applied to oscillatory single-phase solutions of the Korteweg -de Vries (KdV...nonlinear oscillatory waves of the Korteweg - deVries type, Stud. Appl. Math., 78 (1988), pp. 73-90. [5] F. J. Bourland and R. Haberman, The slowly varying
Orbital stability of periodic traveling-wave solutions for the log-KdV equation
NASA Astrophysics Data System (ADS)
Natali, Fábio; Pastor, Ademir; Cristófani, Fabrício
2017-09-01
In this paper we establish the orbital stability of periodic waves related to the logarithmic Korteweg-de Vries equation. Our motivation is inspired in the recent work [3], in which the authors established the well-posedness and the linear stability of Gaussian solitary waves. By using the approach put forward recently in [20] to construct a smooth branch of periodic waves as well as to get the spectral properties of the associated linearized operator, we apply the abstract theories in [13] and [25] to deduce the orbital stability of the periodic traveling waves in the energy space.
Numerical simulation for treatment of dispersive shallow water waves with Rosenau-KdV equation
NASA Astrophysics Data System (ADS)
Ak, Turgut; Battal Gazi Karakoc, S.; Triki, Houria
2016-10-01
In this paper, numerical solutions for the Rosenau-Korteweg-de Vries equation are studied by using the subdomain method based on the sextic B-spline basis functions. Numerical results for five test problems including the motion of single solitary wave, interaction of two and three well-separated solitary waves of different amplitudes, evolution of solitons with Gaussian and undular bore initial conditions are obtained. Stability and a priori error estimate of the scheme are discussed. A comparison of the values of the obtained invariants and error norms for single solitary wave with earlier results is also made. The results show that the present method is efficient and reliable.
NASA Astrophysics Data System (ADS)
Fedotov, I. A.; Polyanin, A. D.
2011-09-01
Broad classes of nonlinear equations of mathematical physics are described that admit order reduction by applying the von Mises transformation (with the unknown function used as a new independent variable and with a suitable partial derivative used as a new dependent variable) and by applying the Crocco transformation (with the first and second partial derivatives used as new independent and dependent variables, respectively). Associated Bäcklund transformations are constructed that connect evolution equations of general form (their special cases include Burgers, Korteweg-de Vries, and Harry Dym type equations and many other nonlinear equations of mathematical physics). Transformations are indicated that reduce the order of hydrodynamic-type equations of higher orders. The generalized Calogero equation and a number of other new integrable nonlinear equations, reducible to linear equations, are considered.
Multiple scales analysis and travelling wave solutions for KdV type nonlinear evolution equations
NASA Astrophysics Data System (ADS)
Ayhan, Burcu; Ozer, M. Naci; Bekir, Ahmet
2017-01-01
Nonlinear evolution equations are the mathematical models of problems that arise in many field of science. These equations has become an important field of study in applied mathematics in recent years. We apply exact solution methods and multiple scale method which is known as a perturbation method to nonlinear evolution equations. Using exact solution methods we get travelling wave solutions expressed by hyperbolic functions, trigonometric functions and rational functions. Also we derive Nonlinear Schrödinger (NLS) type equations from Korteweg-de Vries (KdV) type nonlinear evolution equations and we get approximate solutions for KdV type equations using multiple scale method. The proposed methods are direct and effective and can be used for many nonlinear evolution equations. It is shown that these methods provide a powerful mathematical tool to solve nonlinear evolution equations in mathematical physics.
Chiral smectic-A and smectic-C phases with de Vries characteristics
NASA Astrophysics Data System (ADS)
Yadav, Neelam; Panov, V. P.; Swaminathan, V.; Sreenilayam, S. P.; Vij, J. K.; Perova, T. S.; Dhar, R.; Panov, A.; Rodriguez-Lojo, D.; Stevenson, P. J.
2017-06-01
Infrared and dielectric spectroscopic techniques are used to investigate the characteristics of two chiral smectics, namely, 1,1,3,3,5,5,5-heptamethyltrisiloxane 1-[4'-(undecyl-1-oxy)-4-biphenyl(S,S)-2-chloro-3-methylpentanoate] (MS i3M R11 ) and tricarbosilane-hexyloxy-benzoic acid (S)-4'-(1-methyl-hexyloxy)-3'-nitro-biphenyl-4-yl ester (W599). The orientational features and the field dependencies of the apparent tilt angle and the dichroic ratio for homogeneous planar-aligned samples were calculated from the absorbance profiles obtained at different temperatures especially in the smectic-A* phase of these liquid crystals. The dichroic ratios of the C-C phenyl ring stretching vibrations were considered for the determination of the tilt angle at different temperatures and different voltages. The low values of the order parameter obtained with and without an electric field applied across the cell in the Sm -A* phase for both smectics are consistent with the de Vries concept. The generalized Langevin-Debye model introduced in the literature for explaining the electro-optical response has been applied to the results from infrared spectroscopy. The results show that the dipole moment of the tilt-correlated domain diverges as the transition temperature from Sm -A* to Sm -C* is approached. The Debye-Langevin model is found to be extremely effective in confirming some of the conclusions of the de Vries chiral smectics and gives additional results on the order parameter and the dichroic ratio as a function of the field across the cell. Dielectric spectroscopy finds large dipolar fluctuations in the Sm -A* phase for both compounds and again these confirm their de Vries behavior.
NASA Astrophysics Data System (ADS)
Stamhuis, Ida H.
2015-01-01
Eleven years before the `rediscovery' in 1900 of Mendel's work, Hugo De Vries published his theory of heredity. He expected his theory to become a big success, but it was not well-received. To find supporting evidence for this theory De Vries started an extensive research program. Because of the parallels of his ideas with the Mendelian laws and because of his use of statistics, he became one of the rediscoverers. However, the Mendelian laws, which soon became the foundation of a new discipline of genetics, presented a problem. De Vries was the only one of the early Mendelians who had developed his own theory of heredity. His theory could not be brought in line with the Mendelian laws. But because his original theory was still very dear to him, something important was at stake and he was unwilling to adapt his ideas to the new situation. He belittled the importance of the Mendelian laws and ended up on the sidelines.
ERIC Educational Resources Information Center
Stamhuis, Ida H.
2015-01-01
Eleven years before the "rediscovery" in 1900 of Mendel's work, Hugo De Vries published his theory of heredity. He expected his theory to become a big success, but it was not well-received. To find supporting evidence for this theory De Vries started an extensive research program. Because of the parallels of his ideas with the…
ERIC Educational Resources Information Center
Stamhuis, Ida H.
2015-01-01
Eleven years before the "rediscovery" in 1900 of Mendel's work, Hugo De Vries published his theory of heredity. He expected his theory to become a big success, but it was not well-received. To find supporting evidence for this theory De Vries started an extensive research program. Because of the parallels of his ideas with the…
Integrable nonlinear evolution partial differential equations in 4 + 2 and 3 + 1 dimensions.
Fokas, A S
2006-05-19
The derivation and solution of integrable nonlinear evolution partial differential equations in three spatial dimensions has been the holy grail in the field of integrability since the late 1970s. The celebrated Korteweg-de Vries and nonlinear Schrödinger equations, as well as the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. Do there exist integrable analogs of these equations in three spatial dimensions? In what follows, I present a positive answer to this question. In particular, I first present integrable generalizations of the KP and DS equations, which are formulated in four spatial dimensions and which have the novelty that they involve complex time. I then impose the requirement of real time, which implies a reduction to three spatial dimensions. I also present a method of solution.
Whitham modulation theory for the Kadomtsev- Petviashvili equation.
Ablowitz, Mark J; Biondini, Gino; Wang, Qiao
2017-08-01
The genus-1 Kadomtsev-Petviashvili (KP)-Whitham system is derived for both variants of the KP equation; namely the KPI and KPII equations. The basic properties of the KP-Whitham system, including symmetries, exact reductions and its possible complete integrability, together with the appropriate generalization of the one-dimensional Riemann problem for the Korteweg-de Vries equation are discussed. Finally, the KP-Whitham system is used to study the linear stability properties of the genus-1 solutions of the KPI and KPII equations; it is shown that all genus-1 solutions of KPI are linearly unstable, while all genus-1 solutions of KPII are linearly stable within the context of Whitham theory.
Anomalous temperature dependence of layer spacing of de Vries liquid crystals: Compensation model
Merkel, K.; Kocot, A.; Vij, J. K.; Stevenson, P. J.; Panov, A.; Rodriguez, D.
2016-06-13
Smectic liquid crystals that exhibit temperature independent layer thickness offer technological advantages for their use in displays and photonic devices. The dependence of the layer spacing in SmA and SmC phases of de Vries liquid crystals is found to exhibit distinct features. On entering the SmC phase, the layer thickness initially decreases below SmA to SmC (T{sub A–C}) transition temperature but increases anomalously with reducing temperature despite the molecular tilt increasing. This anomalous observation is being explained quantitatively. Results of IR spectroscopy show that layer shrinkage is caused by tilt of the mesogen's rigid core, whereas the expansion is caused by the chains getting more ordered with reducing temperature. This mutual compensation arising from molecular fragments contributing to the layer thickness differs from the previous models. The orientational order parameter of the rigid core of the mesogen provides direct evidence for de Vries cone model in the SmA phase for the two compounds investigated.
The reactions on Hugo de Vries's Intracellular pangenesis: the discussion with August Weismann.
Stamhuis, Ida H
2003-01-01
In 1889 Hugo de Vries published Intracellular Pangenesis in which he formulated his ideas on heredity. The expectations of the impression these ideas would make did not come true and publication was negated or reviewed critically. From the reactions of his Dutch colleagues and the discussion with the famous German zoologist August Weissmann we conclude that the assertion that each cell contains all hereditary material was controversial and even more the claim that characters are inherited independently of each other. De Vries felt that he had to convince his colleagues of the validity of his theory by providing experimental evidence. He established an important research program which resulted in the rediscovery of Mendal's laws and the publication of The Mutation Theory. This article also illustrates some phenomena that go beyond an interesting episode in the development of theories of heredity. It shows that criticism from colleagues can move a researcher so deeply that he feels compelled to set up an extensive research program. Moreover it illustrates that it is not unusual that a creative scientist is only partially willing to take criticism on his theories into account. Last but not least it demonstrates that common opinion on the validity of specific arguments may change in the course of time.
The zero dispersion limits of nonlinear wave equations
Tso, T.
1992-01-01
In chapter 2 the author uses functional analytic methods and conservation laws to solve the initial-value problem for the Korteweg-de Vries equation, the Benjamin-Bona-Mahony equation, and the nonlinear Schroedinger equation for initial data that satisfy some suitable conditions. In chapter 3 the energy estimates are used to show that the strong convergence of the family of the solutions of the KdV equation obtained in chapter 2 in H[sup 3](R) as [epsilon] [yields] 0; also, it is shown that the strong L[sup 2](R)-limit of the solutions of the BBM equation as [epsilon] [yields] 0 before a critical time. In chapter 4 the author uses the Whitham modulation theory and averaging method to find the 2[pi]-periodic solutions and the modulation equations of the KdV equation, the BBM equation, the Klein-Gordon equation, the NLS equation, the mKdV equation, and the P-system. It is shown that the modulation equations of the KdV equation, the K-G equation, the NLS equation, and the mKdV equation are hyperbolic but those of the BBM equation and the P-system are not hyperbolic. Also, the relations are studied of the KdV equation and the mKdV equation. Finally, the author studies the complex mKdV equation to compare with the NLS equation, and then study the complex gKdV equation.
Darboux Transformations and N-soliton Solutions of Two (2+1)-Dimensional Nonlinear Equations
NASA Astrophysics Data System (ADS)
Wang, Xin; Chen, Yong
2014-04-01
Two Darboux transformations of the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawaka (CDGKS) equation and (2+1)-dimensional modified Korteweg-de Vries (mKdV) equation are constructed through the Darboux matrix method, respectively. N-soliton solutions of these two equations are presented by applying the Darboux transformations N times. The right-going bright single-soliton solution and interactions of two and three-soliton overtaking collisions of the (2+1)-dimensional CDGKS equation are studied. By choosing different seed solutions, the right-going bright and left-going dark single-soliton solutions, the interactions of two and three-soliton overtaking collisions, and kink soliton solutions of the (2+1)-dimensional mKdV equation are investigated. The results can be used to illustrate the interactions of water waves in shallow water.
Evolution of higher order nonlinear equation for the dust ion-acoustic waves in nonextensive plasma
Yasmin, S.; Asaduzzaman, M.; Mamun, A. A.
2012-10-15
There are three different types of nonlinear equations, namely, Korteweg-de Vries (K-dV), modified K-dV (mK-dV), and mixed modified K-dV (mixed mK-dV) equations, for the nonlinear propagation of the dust ion-acoustic (DIA) waves. The effects of electron nonextensivity on DIA solitary waves propagating in a dusty plasma (containing negatively charged stationary dust, inertial ions, and nonextensive q distributed electrons) are examined by solving these nonlinear equations. The basic features of mixed mK-dV (higher order nonlinear equation) solitons are found to exist beyond the K-dV limit. The properties of mK-dV solitons are compared with those of mixed mK-dV solitons. It is found that both positive and negative solitons are obtained depending on the q (nonextensive parameter).
Emamuddin, M.; Yasmin, S.; Mamun, A. A.
2013-04-15
The nonlinear propagation of dust-acoustic waves in a dusty plasma whose constituents are negatively charged dust, Maxwellian ions with two distinct temperatures, and electrons following q-nonextensive distribution, is investigated by deriving a number of nonlinear equations, namely, the Korteweg-de-Vries (K-dV), the modified Korteweg-de-Vries (mK-dV), and the Gardner equations. The basic characteristics of the hump (positive potential) and dip (negative potential) shaped dust-acoustic (DA) Gardner solitons are found to exist beyond the K-dV limit. The effects of two temperature ions and electron nonextensivity on the basic features of DA K-dV, mK-dV, and Gardner solitons are also examined. It has been observed that the DA Gardner solitons exhibit negative (positive) solitons for qq{sub c}) (where q{sub c} is the critical value of the nonextensive parameter q). The implications of our results in understanding the localized nonlinear electrostatic perturbations existing in stellar polytropes, quark-gluon plasma, protoneutron stars, etc. (where ions with different temperatures and nonextensive electrons exist) are also briefly addressed.
NASA Astrophysics Data System (ADS)
King, R. B.
1986-04-01
This paper presents a simple way of classifying higher-order differential equations based on the requirements of the Painlevé property, i.e., the presence of no movable critical points. The fundamental building blocks for such equations may be generated by strongly self-dominant differential equations of the type (∂/∂x)nu =(∂/∂xm)[u(m-n+p)/p] in which m and n are positive integers and p is a negative integer. Such differential equations having both a constant degree d and a constant value of the difference n-m form a Painlevé chain; however, only three of the many possible Painlevé chains can have the Painlevé property. Among the three Painlevé chains that can have the Painlevé property, one contains the Burgers' equation; another contains the dominant terms of the first Painlevé transcendent, the isospectral Korteweg-de Vries equation, and the isospectral Boussinesq equation; and the third contains the dominant terms of the second Painlevé transcendent and the isospectral modified (cubic) Korteweg-de Vries equation. Differential equations of the same order and having the same value of the quotient (n-m)/(d-1) can be mixed to generate a new hybrid differential equation. In such cases a hybrid can have the Painlevé property even if only one of its components has the Painlevé property. Such hybridization processes can be used to generate the various fifth-order evolution equations of interest, namely the Caudrey-Dodd-Gibbon, Kuperschmidt, and Morris equations.
Self-focusing and modulational analysis for nonlinear Schroedinger equations
Weinsten, M.I.
1982-01-01
For the initial-value problem (IVP) for the nonlinear Schroedinger equation, a sufficient condition for the existence of a unique global solution of the IVP is found. The condition is derived by solving a variational problem to obtain the best constant for a classical interpolation estimate of Nirenberg and Gagliardo. A systematic analysis of the singular structure is presented here for the first time. Methods apply to the general critical case. Linear modulational stability of the ground state relative to small perturbations in NLS and/or the initial data is established in the subcritical case. A sufficient condition for the existence of a unique global solution of a generalized Korteweg-de Vries equation is obtained in terms of the solitary (traveling) wave solution.
NASA Astrophysics Data System (ADS)
Wang, Ya-Le; Gao, Yi-Tian; Jia, Shu-Liang; Lan, Zhong-Zhou; Deng, Gao-Fu; Su, Jing-Jing
2017-01-01
Under investigation in this paper is a (2+1)-dimensional generalized variable-coefficient shallow water wave equation which can be reduced to several integrable equations, such as the Korteweg-de Vries (KdV) equation and the Calogero-Bogoyavlenskii-Schiff (CBS) equation. Bilinear forms, Bäcklund transformation, Lax pair and infinite conservation laws are derived based on the binary Bell polynomials. N-soliton solutions are constructed via the Hirota method. Propagation and interaction of the solitons are illustrated graphically: (i) variable coefficients affect the shape of the N-soliton interaction in the scaled space and time coordinates; (ii) positions of the solitons depend on the sign of wave numbers after each interaction; (iii) interaction of the solitons is elastic, i.e. the amplitude, velocity and shape of each soliton remain invariant after each interaction except for a phase shift.
NASA Astrophysics Data System (ADS)
Saha Ray, S.
2013-12-01
In this paper, the modified fractional reduced differential transform method (MFRDTM) has been proposed and it is implemented for solving fractional KdV (Korteweg-de Vries) equations. The fractional derivatives are described in the Caputo sense. In this paper, the reduced differential transform method is modified to be easily employed to solve wide kinds of nonlinear fractional differential equations. In this new approach, the nonlinear term is replaced by its Adomian polynomials. Thus the nonlinear initial-value problem can be easily solved with less computational effort. In order to show the power and effectiveness of the present modified method and to illustrate the pertinent features of the solutions, several fractional KdV equations with different types of nonlinearities are considered. The results reveal that the proposed method is very effective and simple for obtaining approximate solutions of fractional KdV equations.
Pretend model of traveling wave solution of two-dimensional K-dV equation
NASA Astrophysics Data System (ADS)
Karim, Md Rezaul; Alim, Md Abdul; Andallah, Laek Sazzad
2013-11-01
Traveling wave resolution of Korteweg-de Vries (K-dV) solitary and numerical estimation of analytic solutions have been studied in this paper for imaginary concept. Pretend model of traveling wave deals with giant waves or series of waves created by an undersea earthquake, volcanic eruption or landslide. The concept of traveling wave is frequently used by mariners and in coastal, ocean and naval engineering. We have found some exact traveling wave solutions with relevant physical parameters using new auxiliary equation method introduced by Pang et al. (Appl. Math. Mech-Engl. Ed 31(7):929-936, 2010). We have solved the imaginary part of exact traveling wave equations analytically, and numerical results of time-dependent wave solutions have been presented graphically. This procedure has a potential to be used in more complex system for other types of K-dV equations.
Key words: Nonlinear Differential-Difference Equations; Exp-Function Method; N-Soliton Solutions
NASA Astrophysics Data System (ADS)
Dehghan, Mehdi; Manafian, Jalil; Saadatmandi, Abbas
2010-11-01
In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the homotopy analysis method for partial differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.
Origin of weak layer contraction in de Vries smectic liquid crystals
NASA Astrophysics Data System (ADS)
Agra-Kooijman, Dena M.; Yoon, HyungGuen; Dey, Sonal; Kumar, Satyendra
2014-03-01
Structural investigations of the de Vries smectic-A (SmA) and smectic-C (SmC) phases of four mesogens containing a trisiloxane end segment reveal a linear molecular conformation in the SmA phase and a bent conformation resembling a hockey stick in the SmC phase. The siloxane and the hydrocarbon parts of the molecule tilt at different angles relative to the smectic layer normal and are oriented along different directions. For the compounds investigated, the shape of orientational distribution function (ODF) is found to be sugarloaf shaped and not the widely expected volcano like with positive orientational order parameters: ⟨P2⟩ = 0.53-0.78, ⟨P4⟩ = 0.14-0.45, and ⟨P6⟩˜0.10. The increase in the effective molecular length, and consequently in the smectic layer spacing caused by reduced fluctuations and the corresponding narrowing of the ODF, counteracts the effect of molecular tilt and significantly reduces the SmC layer contraction. Maximum tilt of the hydrocarbon part of the molecule lies between approximately 18° and 25° and between 6° and 12° for the siloxane part. The critical exponent of the tilt order parameter, β˜0.25, is in agreement with tricritical behavior at the SmA-SmC transition for two compounds and has lower value for first-order transition in the other compounds with finite enthalpy of transition.
Liu, Ju; Gomez, Hector; Landis, Chad M.
2013-09-01
We propose a new methodology for the numerical solution of the isothermal Navier–Stokes–Korteweg equations. Our methodology is based on a semi-discrete Galerkin method invoking functional entropy variables, a generalization of classical entropy variables, and a new time integration scheme. We show that the resulting fully discrete scheme is unconditionally stable-in-energy, second-order time-accurate, and mass-conservative. We utilize isogeometric analysis for spatial discretization and verify the aforementioned properties by adopting the method of manufactured solutions and comparing coarse mesh solutions with overkill solutions. Various problems are simulated to show the capability of the method. Our methodology provides a means of constructing unconditionally stable numerical schemes for nonlinear non-convex hyperbolic systems of conservation laws.
Nonlinear disintegration of sine wave in the framework of the Gardner equation
NASA Astrophysics Data System (ADS)
Kurkina, Oxana; Rouvinskaya, Ekaterina; Talipova, Tatiana; Kurkin, Andrey; Pelinovsky, Efim
2016-10-01
Internal tidal wave entering shallow waters transforms into an undular bore and this process can be described in the framework of the Gardner equation (extended version of the Korteweg-de Vries equation with both quadratic and cubic nonlinear terms). Our numerical computations demonstrate the features of undular bore developing for different signs of the cubic nonlinear term. If cubic nonlinear term is negative, and initial wave amplitude is large enough, two undular bores are generated from the two breaking points formed on both crest slopes (within dispersionless Gardner equation). Undular bore consists of one table-top soliton and a group of small soliton-like waves passing through the table-top soliton. If the cubic nonlinear term is positive and again the wave amplitude is large enough, the breaking points appear on crest and trough generating groups of positive and negative soliton-like pulses. This is the main difference with respect to the classic Korteweg-de Vries equation, where the breaking point is single. It is shown also that nonlinear interaction of waves happens similarly to one of scenarios of two-soliton interaction of "exchange" or "overtake" types with a phase shift. If small-amplitude pulses interact with large-amplitude soliton-like pulses, their speed in average is negative in the case when "free" velocity is positive. Nonlinear interaction leads to the generation of higher harmonics and spectrum width increases with amplitude increase independently of the sign of cubic nonlinear term. The breaking asymptotic k 4 / 3 predicted within the dispersionless Gardner equation emerges during the process of undular bore development. The formation of soliton-like perturbations leads to appearance of several spectral peaks which are downshifting with time.
A Haar wavelet collocation method for coupled nonlinear Schrödinger-KdV equations
NASA Astrophysics Data System (ADS)
Oruç, Ömer; Esen, Alaattin; Bulut, Fatih
2016-04-01
In this paper, to obtain accurate numerical solutions of coupled nonlinear Schrödinger-Korteweg-de Vries (KdV) equations a Haar wavelet collocation method is proposed. An explicit time stepping scheme is used for discretization of time derivatives and nonlinear terms that appeared in the equations are linearized by a linearization technique and space derivatives are discretized by Haar wavelets. In order to test the accuracy and reliability of the proposed method L2, L∞ error norms and conserved quantities are used. Also obtained results are compared with previous ones obtained by finite element method, Crank-Nicolson method and radial basis function meshless methods. Error analysis of Haar wavelets is also given.
A numerical dressing method for the nonlinear superposition of solutions of the KdV equation
NASA Astrophysics Data System (ADS)
Trogdon, Thomas; Deconinck, Bernard
2014-01-01
In this paper we present the unification of two existing numerical methods for the construction of solutions of the Korteweg-de Vries (KdV) equation. The first method is used to solve the Cauchy initial-value problem on the line for rapidly decaying initial data. The second method is used to compute finite-genus solutions of the KdV equation. The combination of these numerical methods allows for the computation of exact solutions that are asymptotically (quasi-)periodic finite-gap solutions and are a nonlinear superposition of dispersive, soliton and (quasi-)periodic solutions in the finite (x, t)-plane. Such solutions are referred to as superposition solutions. We compute these solutions accurately for all values of x and t.
Kets de Vries, Manfred F
2004-01-01
Much of the business literature on leadership starts with the assumption that leaders are rational beings. But irrationality is integral to human nature, and inner conflict often contributes to the drive to succeed. Although a number of business scholars have explored the psychology of executives, Manfred F.R Kets de Vries has made the analysis of CEOs his life's work. In this article, Kets de Vries, a psychoanalyst, author, and instead professor, draws on three decades of study to describe the psychological profile of successful CEOs. He explores senior executives' vulnerabilities, which are often intensified by followers' attempts to manipulate their leaders. Leaders, he says, have an uncanny ability to awaken transferential processes--in which people transfer the dynamics of past relationships onto present interactions--among their employees and even in themselves. These processes can present themselves in a number of ways, sometimes negatively. What's more, many top executives, being middle-aged, suffer from depression. Mid-life prompts a reappraisal of career identity, and by the time a leader is a CEO, an existential crisis is often imminent. This can happen with anyone, but the probability is higher with CEOs, and senior executives because so many have devoted themselves exclusively to work. Not all CEOs are psychologically unhealthy, of course. Healthy leaders are talented in self-observation and self-analysis, Kets de Vries says. The best are highly motivated to spend time on self-reflection. Their lives are in balance, they can play, they are creative and inventive, and they have the capacity to be nonconformist. "Those who accept the madness in themselves may be the healthiest leaders of all," he concludes.
A Unified Model for the Evolution of Nonlinear Water Waves.
1982-12-30
Korteweg and deVries , 1895) were set down. In their pioneering nierical study of solutions to the Korteweg and deVries (KdeV) Equation , Zabusky and Kruskal...theories that have been used to study colliding solitary waves in water are the Korteweg - deVries equation - I3n -&. + + Ti 1 aI T 3x ax ax and the...9l. Miles J.W., 1979, On the Korteweg - deVries equation for a gradual’y .ari channel, J. Fluid Mech., 91, 11-190. Peregrine, D.H., 1966,
Prasad, S Krishna; Rao, D S Shankar; Sridevi, S; Lobo, Chethan V; Ratna, B R; Naciri, Jawad; Shashidhar, R
2009-04-10
X-ray, electrical, electro-optical, and dielectric studies in the de Vries smectic A (SmA) phase of organosiloxane derivatives exhibit features surprisingly different from that of a conventional SmA phase. The switching data show a double peak profile, characteristic of an antiferroelectric (AF) structure. A model with the adjacent smectic layers having an AF-like arrangement and no global tilt correlation is proposed. Observed in molecules with differential interactions between the two termini, these findings have wide ramifications in understanding the minimum layer shrinkage of such systems.
Xue Jukui
2006-02-15
In a recent paper, V. A. Brazhnyi and V. V. Konoto [Phys. Rev. E 72, 026616 (2005)] investigated the dynamics of vector dark solitons in two-component Bose-Einstein condensates. In the small amplitude limit, they deduced a coupled Korteweg-de Vries equation from the coupled Gross-Pitaevskii equations. They found that two branches of (slow and fast) dark solitons corresponding to the two branches of the sound waves exist. The slow solitons, corresponding to the lower branch of the acoustic wave, appear to be unstable and transform during the evolution into the stable fast solitons (corresponding to the upper branch of the dispersion law). However, our discussion shows that these results are incorrect.
Xue, Ju-Kui
2006-02-01
In a recent paper, V. A. Brazhnyi and V. V. Konoto [Phys. Rev. E 72, 026616 (2005)] investigated the dynamics of vector dark solitons in two-component Bose-Einstein condensates. In the small amplitude limit, they deduced a coupled Korteweg-de Vries equation from the coupled Gross-Pitaevskii equations. They found that two branches of (slow and fast) dark solitons corresponding to the two branches of the sound waves exist. The slow solitons, corresponding to the lower branch of the acoustic wave, appear to be unstable and transform during the evolution into the stable fast solitons (corresponding to the upper branch of the dispersion law). However, our discussion shows that these results are incorrect.
Discrete transparent boundary conditions for the mixed KDV-BBM equation
NASA Astrophysics Data System (ADS)
Besse, Christophe; Noble, Pascal; Sanchez, David
2017-09-01
In this paper, we consider artificial boundary conditions for the linearized mixed Korteweg-de Vries (KDV) and Benjamin-Bona-Mahoney (BBM) equation which models water waves in the small amplitude, large wavelength regime. Continuous (respectively discrete) artificial boundary conditions involve non local operators in time which in turn requires to compute time convolutions and invert the Laplace transform of an analytic function (respectively the Z-transform of an holomorphic function). In this paper, we propose a new, stable and fairly general strategy to carry out this crucial step in the design of transparent boundary conditions. For large time simulations, we also introduce a methodology based on the asymptotic expansion of coefficients involved in exact direct transparent boundary conditions. We illustrate the accuracy of our methods for Gaussian and wave packets initial data.
NASA Astrophysics Data System (ADS)
Lou, Sen-yue
1998-05-01
To study a nonlinear partial differential equation (PDE), the Painleve expansion developed by Weiss, Tabor and Carnevale (WTC) is one of the most powerful methods. In this paper, using any singular manifold, the expansion series in the usual Painleve analysis is shown to be resummable in some different ways. A simple nonstandard truncated expansion with a quite universal reduction function is used for many nonlinear integrable and nonintegrable PDEs such as the Burgers, Korteweg de-Vries (KdV), Kadomtsev-Petviashvli (KP), Caudrey-Dodd-Gibbon-Sawada-Kortera (CDGSK), Nonlinear Schrödinger (NLS), Davey-Stewartson (DS), Broer-Kaup (BK), KdV-Burgers (KdVB), λf4 , sine-Gordon (sG) etc.
Kocot, A; Vij, J K; Perova, T S; Merkel, K; Swaminathan, V; Sreenilayam, S P; Yadav, N; Panov, V P; Stevenson, P J; Panov, A; Rodriguez-Lojo, D
2017-09-07
Two approaches exist in the literature for describing the orientational distribution function (ODF) of the molecular directors in SmA* phase of liquid crystals, though several models are recently proposed in the literature for explaining the de Vries behaviour. These ODFs correspond to either the conventional unimodal arrangements of molecular directors arising from the mean field theory that leads to the broad or sugar-loaf like distribution or to the "diffuse-cone-shaped" type distribution proposed by de Vries. The hypothesis by de Vries provides for a realistic explanation as to how at a molecular level, a first-order SmA* to SmC* transition can occur where the uniform molecular director azimuthal distributions condense to values lying within a narrow range of angles; finally these condense to a single value while at the same time ensuring a little or no concomitant shrinkage in the layer spacing. The azimuthal distribution of the in-layer directors is probed using IR and polarized Raman spectroscopic techniques. The latter allows us to obtain the ODF and the various order parameters for the uniaxial and the biaxial phases. Based on the results of these measurements, we conclude that the "cone-shaped" (or volcano-shaped) de Vries type of distribution can most preferably describe SmA* where "a first-order phase transition from SmA* to SmC*" and a low layer shrinkage can both be easily explained.
NASA Astrophysics Data System (ADS)
Kocot, A.; Vij, J. K.; Perova, T. S.; Merkel, K.; Swaminathan, V.; Sreenilayam, S. P.; Yadav, N.; Panov, V. P.; Stevenson, P. J.; Panov, A.; Rodriguez-Lojo, D.
2017-09-01
Two approaches exist in the literature for describing the orientational distribution function (ODF) of the molecular directors in SmA* phase of liquid crystals, though several models are recently proposed in the literature for explaining the de Vries behaviour. These ODFs correspond to either the conventional unimodal arrangements of molecular directors arising from the mean field theory that leads to the broad or sugar-loaf like distribution or to the "diffuse-cone-shaped" type distribution proposed by de Vries. The hypothesis by de Vries provides for a realistic explanation as to how at a molecular level, a first-order SmA* to SmC* transition can occur where the uniform molecular director azimuthal distributions condense to values lying within a narrow range of angles; finally these condense to a single value while at the same time ensuring a little or no concomitant shrinkage in the layer spacing. The azimuthal distribution of the in-layer directors is probed using IR and polarized Raman spectroscopic techniques. The latter allows us to obtain the ODF and the various order parameters for the uniaxial and the biaxial phases. Based on the results of these measurements, we conclude that the "cone-shaped" (or volcano-shaped) de Vries type of distribution can most preferably describe SmA* where "a first-order phase transition from SmA* to SmC*" and a low layer shrinkage can both be easily explained.
Higher Painlevé transcendents as special solutions of some nonlinear integrable hierarchies
NASA Astrophysics Data System (ADS)
Kudryashov, Nikolay A.
2014-02-01
It is well known that the self-similar solutions of the Korteweg-de Vries equation and the modified Korteweg-de Vries equation are expressed via the solutions of the first and second Painlevé equations. In this paper we solve this problem for all equations from the Korteveg-de Vries, modified Korteweg-de Vries, Kaup-Kupershmidt, Caudrey-Dodd-Gibbon and Fordy-Gibbons hierarchies. We show that the self-similar solutions of equations corresponding to hierarchies mentioned above can be found by means of the general solutions of higher-order Painlevé hierarchies introduced more than ten years ago.
NASA Astrophysics Data System (ADS)
Trogdon, Thomas; Deconinck, Bernard
2014-01-01
All solutions of the Korteweg-de Vries equation that are bounded on the real line are physically relevant, depending on the application area of interest. Usually, both analytical and numerical approaches consider solution profiles that are either spatially localized or (quasi-)periodic. In this paper, we discuss a class of solutions that is a nonlinear superposition of these two cases: their asymptotic state for large |x| is (quasi-)periodic, but they may contain solitons, with or without dispersive tails. Such scenarios might occur in the case of localized perturbations of previously present sea swell, for instance. Such solutions have been discussed from an analytical point of view only recently. We numerically demonstrate different features of these solutions.
Dispersive shock waves in the Kadomtsev-Petviashvili and two dimensional Benjamin-Ono equations
NASA Astrophysics Data System (ADS)
Ablowitz, Mark J.; Demirci, Ali; Ma, Yi-Ping
2016-10-01
Dispersive shock waves (DSWs) in the Kadomtsev-Petviashvili (KP) equation and two dimensional Benjamin-Ono (2DBO) equation are considered using step like initial data along a parabolic front. Employing a parabolic similarity reduction exactly reduces the study of such DSWs in two space one time (2 + 1) dimensions to finding DSW solutions of (1 + 1) dimensional equations. With this ansatz, the KP and 2DBO equations can be exactly reduced to the cylindrical Korteweg-de Vries (cKdV) and cylindrical Benjamin-Ono (cBO) equations, respectively. Whitham modulation equations which describe DSW evolution in the cKdV and cBO equations are derived and Riemann type variables are introduced. DSWs obtained from the numerical solutions of the corresponding Whitham systems and direct numerical simulations of the cKdV and cBO equations are compared with very good agreement obtained. In turn, DSWs obtained from direct numerical simulations of the KP and 2DBO equations are compared with the cKdV and cBO equations, again with good agreement. It is concluded that the (2 + 1) DSW behavior along self similar parabolic fronts can be effectively described by the DSW solutions of the reduced (1 + 1) dimensional equations.
Conservation laws and symmetries of Hunter-Saxton equation: revisited
NASA Astrophysics Data System (ADS)
Tian, Kai; Liu, Q. P.
2016-03-01
Through a reciprocal transformation {{T}0} induced by the conservation law {{\\partial}t}≤ft(ux2\\right)={{\\partial}x}≤ft(2uux2\\right) , the Hunter-Saxton (HS) equation {{u}xt}=2u{{u}2x}+ux2 is shown to possess conserved densities involving arbitrary smooth functions, which have their roots in infinitesimal symmetries of {{w}t}={{w}2} , the counterpart of the HS equation under {{T}0} . Hierarchies of commuting symmetries of the HS equation are studied under appropriate changes of variables initiated by {{T}0} , and two of these are linearized while the other is identical to the hierarchy of commuting symmetries admitted by the potential modified Korteweg-de Vries equation. A fifth order symmetry of the HS equation is endowed with a sixth order hereditary recursion operator, which is proved to have a bi-Hamiltonian factorization, by its connection with the Fordy-Gibbons equation. These results reveal the origin for the rich and remarkable structures of the HS equation and partially answer the questions raised by Wang (2010 Nonlinearity 23 2009).
Data-driven discovery of partial differential equations.
Rudy, Samuel H; Brunton, Steven L; Proctor, Joshua L; Kutz, J Nathan
2017-04-01
We propose a sparse regression method capable of discovering the governing partial differential equation(s) of a given system by time series measurements in the spatial domain. The regression framework relies on sparsity-promoting techniques to select the nonlinear and partial derivative terms of the governing equations that most accurately represent the data, bypassing a combinatorially large search through all possible candidate models. The method balances model complexity and regression accuracy by selecting a parsimonious model via Pareto analysis. Time series measurements can be made in an Eulerian framework, where the sensors are fixed spatially, or in a Lagrangian framework, where the sensors move with the dynamics. The method is computationally efficient, robust, and demonstrated to work on a variety of canonical problems spanning a number of scientific domains including Navier-Stokes, the quantum harmonic oscillator, and the diffusion equation. Moreover, the method is capable of disambiguating between potentially nonunique dynamical terms by using multiple time series taken with different initial data. Thus, for a traveling wave, the method can distinguish between a linear wave equation and the Korteweg-de Vries equation, for instance. The method provides a promising new technique for discovering governing equations and physical laws in parameterized spatiotemporal systems, where first-principles derivations are intractable.
NASA Astrophysics Data System (ADS)
Ge, Hong-Xia; Lai, Ling-Ling; Zheng, Peng-Jun; Cheng, Rong-Jun
2013-12-01
A new continuum traffic flow model is proposed based on an improved car-following model, which takes the driver's forecast effect into consideration. The backward travel problem is overcome by our model and the neutral stability condition of the new model is obtained through the linear stability analysis. Nonlinear analysis shows clearly that the density fluctuation in traffic flow leads to a variety of density waves and the Korteweg-de Vries-Burgers (KdV-Burgers) equation is derived to describe the traffic flow near the neutral stability line. The corresponding solution for traffic density wave is also derived. Finally, the numerical results show that our model can not only reproduce the evolution of small perturbation, but also improve the stability of traffic flow.
Numerical solutions of nonlinear wave equations
Kouri, D.J.; Zhang, D.S.; Wei, G.W.; Konshak, T.; Hoffman, D.K.
1999-01-01
Accurate, stable numerical solutions of the (nonlinear) sine-Gordon equation are obtained with particular consideration of initial conditions that are exponentially close to the phase space homoclinic manifolds. Earlier local, grid-based numerical studies have encountered difficulties, including numerically induced chaos for such initial conditions. The present results are obtained using the recently reported distributed approximating functional method for calculating spatial derivatives to high accuracy and a simple, explicit method for the time evolution. The numerical solutions are chaos-free for the same conditions employed in previous work that encountered chaos. Moreover, stable results that are free of homoclinic-orbit crossing are obtained even when initial conditions are within 10{sup {minus}7} of the phase space separatrix value {pi}. It also is found that the present approach yields extremely accurate solutions for the Korteweg{endash}de Vries and nonlinear Schr{umlt o}dinger equations. Our results support Ablowitz and co-workers{close_quote} conjecture that ensuring high accuracy of spatial derivatives is more important than the use of symplectic time integration schemes for solving solitary wave equations. {copyright} {ital 1999} {ital The American Physical Society}
A new perturbative approach to nonlinear partial differential equations
Bender, C.M.; Boettcher, S. ); Milton, K.A. )
1991-11-01
This paper shows how to solve some nonlinear wave equations as perturbation expansions in powers of a parameter that expresses the degree of nonlinearity. For the case of the Burgers equation {ital u}{sub {ital t}}+{ital uu}{sub {ital x}}={ital u}{sub {ital xx}}, the general nonlinear equation {ital u}{sub {ital t}}+{ital u}{sup {delta}}{ital u}{sub {ital x}}={ital u}{sub {ital xx}} is considered and expanded in powers of {delta}. The coefficients of the {delta} series to sixth order in powers of {delta} is determined and Pade summation is used to evaluate the perturbation series for large values of {delta}. The numerical results are accurate and the method is very general; it applies to other well-studied partial differential equations such as the Korteweg--de Vries equation, {ital u}{sub {ital t}}+{ital uu}{sub {ital x}} ={ital u}{sub {ital xxx}}.
The frequency-dependent electrooptic response of the electroclinic effect in deVries SmA materials
NASA Astrophysics Data System (ADS)
Jones, Christopher D.; Dawin, Ute; Giesselmann, Frank; Clark, Noel; Rudquist, Per
2007-03-01
It is well established that electroclinic switching in standard SmA* materials relates to a reorientation of the molecules in a plane normal to the layers, and thus there is no corresponding change in birefringence due to reorientation about a cone, as is the case in the SmC* phase. When the electrooptic response is then analyzed via lock-in amplifier, the signal at the driving frequency is strong, while the second harmonic response, is non-existent [1]. Using this method we have investigated deVries materials W530 and TSiKN65, and show that there is a frequency-dependent second order response -- implying an electroclinic switching that corresponds to a change in birefringence, suggesting a reorientation of the molecule about a cone. We will present our findings and a model for the type of electroclinic switching that occurs in these two materials. Work supported by NSF MRSEC Grant DMR-0213918 and The Swedish Foundation for Strategic Research 2002/0388. [1] W. Kuczynski, et. al., Ferroelectics, 244, [491]/191, (2000)
On the orbital stability of Gaussian solitary waves in the log-KdV equation
NASA Astrophysics Data System (ADS)
Carles, Rémi; Pelinovsky, Dmitry
2014-12-01
We consider the logarithmic Korteweg-de Vries (log-KdV) equation, which models solitary waves in anharmonic chains with Hertzian interaction forces. By using an approximating sequence of global solutions of the regularized generalized KdV equation in H^1({R}) with conserved L2 norm and energy, we construct a weak global solution of the log-KdV equation in a subset of H^1({R}) . This construction yields conditional orbital stability of Gaussian solitary waves of the log-KdV equation, provided that uniqueness and continuous dependence of the constructed solution holds. Furthermore, we study the linearized log-KdV equation at the Gaussian solitary wave and prove that the associated linearized operator has a purely discrete spectrum consisting of simple purely imaginary eigenvalues in addition to the double zero eigenvalue. The eigenfunctions, however, do not decay like Gaussian functions but have algebraic decay. Using numerical approximations, we show that the Gaussian initial data do not spread out but produce visible radiation at the left slope of the Gaussian-like pulse in the time evolution of the linearized log-KdV equation.
Modulational Instability and Rogue Waves in Shallow Water Models
NASA Astrophysics Data System (ADS)
Grimshaw, R.; Chow, K. W.; Chan, H. N.
It is now well known that the focussing nonlinear Schrödinger equation allows plane waves to be modulationally unstable, and at the same time supports breather solutions which are often invoked as models for rogue waves. This suggests a direct connection between modulation instability and the existence of rogue waves. In this chapter we review this connection for a suite of long wave models, such as the Korteweg-de Vries equation, the extended Korteweg-de Vries (Gardner) equation, often used to describe surface and internal waves in shallow water, a Boussinesq equation and, also a coupled set of Korteweg-de Vries equations.
NASA Astrophysics Data System (ADS)
El-Tantawy, S. A.
2016-05-01
We examine the likelihood of the ion-acoustic rogue waves propagation in a non-Maxwellian electronegative plasma in the framework of the family of the Korteweg-de Vries (KdV) equations (KdV/modified KdV/Extended KdV equation). For this purpose, we use the reductive perturbation technique to carry out this study. It is known that the family of the KdV equations have solutions of distinct structures such as solitons, shocks, kinks, cnoidal waves, etc. However, the dynamics of the nonlinear rogue waves is governed by the nonlinear Schrödinger equation (NLSE). Thus, the family of the KdV equations is transformed to their corresponding NLSE developing a weakly nonlinear wave packets. We show the possible region for the existence of the rogue waves and define it precisely for typical parameters of space plasmas. We investigate numerically the effects of relevant physical parameters, namely, the negative ion relative concentration, the nonthermal parameter, and the mass ratio on the propagation of the rogue waves profile. The present study should be helpful in understanding the salient features of the nonlinear structures such as, ion-acoustic solitary waves, shock waves, and rogue waves in space and in laboratory plasma where two distinct groups of ions, i.e. positive and negative ions, and non-Maxwellian (nonthermal) electrons are present.
INVITED ARTICLE: The second Painlevé equation, its hierarchy and associated special polynomials
NASA Astrophysics Data System (ADS)
Clarkson, Peter A.; Mansfield, Elizabeth L.
2003-05-01
In this paper we are concerned with hierarchies of rational solutions and associated polynomials for the second Painlevé equation (PII) and the equations in the PII hierarchy which is derived from the modified Korteweg-de Vries hierarchy. These rational solutions of PII are expressible as the logarithmic derivative of special polynomials, the Yablonskii-Vorob'ev polynomials. The structure of the roots of these Yablonskii-Vorob'ev polynomials is studied and it is shown that these have a highly regular triangular structure. Further, the properties of the Yablonskii-Vorob'ev polynomials are compared and contrasted with those of classical orthogonal polynomials. We derive the special polynomials for the second and third equations of the PII hierarchy and give a representation of the associated rational solutions in the form of determinants through Schur functions. Additionally the analogous special polynomials associated with rational solutions and representation in the form of determinants are conjectured for higher equations in the PII hierarchy. The roots of these special polynomials associated with rational solutions for the equations of the PII hierarchy also have a highly regular structure.
Wave-packet formation at the zero-dispersion point in the Gardner-Ostrovsky equation
NASA Astrophysics Data System (ADS)
Whitfield, A. J.; Johnson, E. R.
2015-05-01
The long-time effect of weak rotation on an internal solitary wave is the decay into inertia-gravity waves and the eventual emergence of a coherent, steadily propagating, nonlinear wave packet. There is currently no entirely satisfactory explanation as to why these wave packets form. Here the initial value problem is considered within the context of the Gardner-Ostrovsky, or rotation-modified extended Korteweg-de Vries, equation. The linear Gardner-Ostrovsky equation has maximum group velocity at a critical wave number, often called the zero-dispersion point. It is found here that a nonlinear splitting of the wave-number spectrum at the zero-dispersion point, where energy is shifted into the modulationally unstable regime of the Gardner-Ostrovsky equation, is responsible for the wave-packet formation. Numerical comparisons of the decay of a solitary wave in the Gardner-Ostrovsky equation and a derived nonlinear Schrödinger equation at the zero-dispersion point are used to confirm the spectral splitting.
Wave-packet formation at the zero-dispersion point in the Gardner-Ostrovsky equation.
Whitfield, A J; Johnson, E R
2015-05-01
The long-time effect of weak rotation on an internal solitary wave is the decay into inertia-gravity waves and the eventual emergence of a coherent, steadily propagating, nonlinear wave packet. There is currently no entirely satisfactory explanation as to why these wave packets form. Here the initial value problem is considered within the context of the Gardner-Ostrovsky, or rotation-modified extended Korteweg-de Vries, equation. The linear Gardner-Ostrovsky equation has maximum group velocity at a critical wave number, often called the zero-dispersion point. It is found here that a nonlinear splitting of the wave-number spectrum at the zero-dispersion point, where energy is shifted into the modulationally unstable regime of the Gardner-Ostrovsky equation, is responsible for the wave-packet formation. Numerical comparisons of the decay of a solitary wave in the Gardner-Ostrovsky equation and a derived nonlinear Schrödinger equation at the zero-dispersion point are used to confirm the spectral splitting.
Aspects of Integrability in One and Several Dimensions,
1986-01-01
Korteweg - deVries (KdV) equation qt = q - 6qq x q = q(x,t), the recursion operator I is D 4q - 2q Dl. where 0 3x’ (D1 f)(x) f/xf( )dE. If 6 is the...70 It is also well known that the Harry-Dym 6 2 equation , can be mapped to a modified Korteweg - deVries (MKdV) equation via an extended...Petviashvili (a two dimensional analogue of the Korteweg - deVries ) and the Davey-Stewartson (a two dimensional analogue of the nonlinear Schr6dinger
Variable rate irrigation (VRI)
USDA-ARS?s Scientific Manuscript database
Variable rate irrigation (VRI) technology is now offered by all major manufacturers of moving irrigation systems, mostly on center pivot irrigation systems. Variable irrigation depths may be controlled by sector only, in which case only the speed of the irrigation lateral is regulated. Or, variable ...
Atmospheric Fluctuations Which Lead to Trackable Radar Signals in the Marine Boundary Layer.
1981-07-01
R. M., (1967); "Method for Solving the Korteweg - deVries Equation ", Phys. Rev. Lett. 19, 1095-1097. Gardner, C. S., Greene, J. M., Kruskal, M. D...long waves of small amplitude propagate according to Eq. (3.12) on a short time scale, and according to the Korteweg - deVries equation , ut + 6uux + U...and Miura, R. M., (1974); " Korteweg - deVries Equation and Generalization. VI. Methods for Exact Solution", Comm. Pure Appl. Math. 27, 97-133. Gedzelman
NASA Astrophysics Data System (ADS)
Sreenilayam, S. P.; Agra-Kooijman, D. M.; Panov, V. P.; Swaminathan, V.; Vij, J. K.; Panarin, Yu. P.; Kocot, A.; Panov, A.; Rodriguez-Lojo, D.; Stevenson, P. J.; Fisch, Michael R.; Kumar, Satyendra
2017-03-01
A heptamethyltrisiloxane liquid crystal (LC) exhibiting I -Sm A*-Sm C* phases has been characterized by calorimetry, polarizing microscopy, x-ray diffraction, electro-optics, and dielectric spectroscopy. Observations of a large electroclinic effect, a large increase in the birefringence (Δ n ) with electric field, a low shrinkage in the layer thickness (˜1.75%) at 20 °C below the Sm A*-Sm C* transition, and low values of the reduction factor (˜0.40) suggest that the Sm A* phase in this material is of the de Vries type. The reduction factor is a measure of the layer shrinkage in the Sm C* phase and it should be zero for an ideal de Vries. Moreover, a decrease in the magnitude of Δ n with decreasing temperature indicates the presence of the temperature-dependent tilt angle in the Sm A* phase. The electro-optic behavior is explained by the generalized Langevin-Debye model as given by Shen et al. [Y. Shen et al., Phys. Rev. E 88, 062504 (2013), 10.1103/PhysRevE.88.062504]. The soft-mode dielectric relaxation strength shows a critical behavior when the system goes from the Sm A* to the Sm C* phase.
NASA Astrophysics Data System (ADS)
Sanchez-Castillo, A.; Osipov, M. A.; Jagiella, S.; Nguyen, Z. H.; Kašpar, M.; Hamplovă, V.; Maclennan, J.; Giesselmann, F.
2012-06-01
The orientational order parameters
Semiclassical Soliton Ensembles for the Three-Wave Resonant Interaction Equations
NASA Astrophysics Data System (ADS)
Buckingham, R. J.; Jenkins, R. M.; Miller, P. D.
2017-09-01
The three-wave resonant interaction equations are a non-dispersive system of partial differential equations with quadratic coupling describing the time evolution of the complex amplitudes of three resonant wave modes. Collisions of wave packets induce energy transfer between different modes via pumping and decay. We analyze the collision of two or three packets in the semiclassical limit by applying the inverse-scattering transform. Using WKB analysis, we construct an associated semiclassical soliton ensemble, a family of reflectionless solutions defined through their scattering data, intended to accurately approximate the initial data in the semiclassical limit. The map from the initial packets to the soliton ensemble is explicit and amenable to asymptotic and numerical analysis. Plots of the soliton ensembles indicate the space-time plane is partitioned into regions containing either quiescent, slowly varying, or rapidly oscillatory waves. This behavior resembles the well-known generation of dispersive shock waves in equations such as the Korteweg-de Vries and nonlinear Schrödinger equations, although the physical mechanism must be different in the absence of dispersion.
Nonlinear Waves and Inverse Scattering
1989-01-01
5) Numerical Simulation of the Modified Korteweg - deVries Equation , Thiab R. Taha and M.J. Ablowitz, 6th International Symposium on Computer Methods in... solved by the IST method. . Numerically Induced Chaos) /i We have been studying a class of non ’linear equations and their discrete approximations...Certain Nonlinear Evolution Equations IV, Numerical, Modified Korteweg -de Vries Equation , T.R. Taha and M.J. Ablowitz, J. Comp. Physics, Vol. 77, No
Brazhnyi, V.A.; Konotop, V.V.
2005-08-01
The dynamics of vector dark solitons in two-component Bose-Einstein condensates is studied within the framework of coupled one-dimensional nonlinear Schroedinger (NLS) equations. We consider the small-amplitude limit in which the coupled NLS equations are reduced to coupled Korteweg-de Vries (KdV) equations. For a specific choice of the parameters the obtained coupled KdV equations are exactly integrable. We find that there exist two branches of (slow and fast) dark solitons corresponding to the two branches of the sound waves. Slow solitons, corresponding to the lower branch of the acoustic wave, appear to be unstable and transform during the evolution into stable fast solitons (corresponding to the upper branch of the dispersion law). Vector dark solitons of arbitrary depths are studied numerically. It is shown that effectively different parabolic traps, to which the two components are subjected, cause an instability of the solitons, leading to a splitting of their components and subsequent decay. A simple phenomenological theory, describing the oscillations of vector dark solitons in a magnetic trap, is proposed.
Analytical integrability and physical solutions of d-KdV equation
Karmakar, P.K.; Dwivedi, C.B.
2006-03-15
A new idea of electron inertia-induced ion sound wave excitation for transonic plasma equilibrium has already been reported. In such unstable plasma equilibrium, a linear source driven Korteweg-de Vries (d-KdV) equation describes the nonlinear ion sound wave propagation behavior. By numerical techniques, two distinct classes of solution (soliton and oscillatory shocklike structures) are obtained. Present contribution deals with the systematic methodological efforts to find out its (d-KdV) analytical solutions. As a first step, we apply the Painleve method to test whether the derived d-KdV equation is analytically integrable or not. We find that the derived d-KdV equation is indeed analytically integrable since it satisfies Painleve property. Hirota's bilinearization method and the modified sine-Gordon method (also termed as sine-cosine method) are used to derive the analytical results. Perturbative technique is also applied to find out quasistationary solutions. A few graphical plots are provided to offer a glimpse of the structural profiles obtained by different methods applied. It is conjectured that these solutions may open a new scope of acoustic spectroscopy in plasma hydrodynamics.
Mathematical Problems of Nonlinear Wave Propagation and of Waves in Heterogeneous Media.
1986-10-22
Landau type by examining secondary bifurcation. Professor Venakides has completed two papers on the Korteweg -de Vries equation . One shows how step-like...Venakldes The Zero Dispersion Limit of the Korteweg -de Vries Equation for Initial Potentials with Non-trivial Reflection Coefficient Pub: Comm. Pure...Keller The oic dimensional Schr6dingcr equation is solved asymptotically for scattering of a particle by a potential barrier and for botnid staLes of a
Initial-value problem for the Gardner equation applied to nonlinear internal waves
NASA Astrophysics Data System (ADS)
Rouvinskaya, Ekaterina; Kurkina, Oxana; Kurkin, Andrey; Talipova, Tatiana; Pelinovsky, Efim
2017-04-01
The Gardner equation is a fundamental mathematical model for the description of weakly nonlinear weakly dispersive internal waves, when cubic nonlinearity cannot be neglected. Within this model coefficients of quadratic and cubic nonlinearity can both be positive as well as negative, depending on background conditions of the medium, where waves propagate (sea water density stratification, shear flow profile) [Rouvinskaya et al., 2014, Kurkina et al., 2011, 2015]. For the investigation of weakly dispersive behavior in the framework of nondimensional Gardner equation with fixed (positive) sign of quadratic nonlinearity and positive or negative cubic nonlinearity {eq1} partial η/partial t+6η( {1± η} )partial η/partial x+partial ^3η/partial x^3=0, } the series of numerical experiments of initial-value problem was carried out for evolution of a bell-shaped impulse of negative polarity (opposite to the sign of quadratic nonlinear coefficient): {eq2} η(x,t=0)=-asech2 ( {x/x0 } ), for which amplitude a and width x0 was varied. Similar initial-value problem was considered in the paper [Trillo et al., 2016] for the Korteweg - de Vries equation. For the Gardner equation with different signs of cubic nonlinearity the initial-value problem for piece-wise constant initial condition was considered in detail in [Grimshaw et al., 2002, 2010]. It is widely known, for example, [Pelinovsky et al., 2007], that the Gardner equation (1) with negative cubic nonlinearity has a family of classic solitary wave solutions with only positive polarity,and with limiting amplitude equal to 1. Therefore evolution of impulses (2) of negative polarity (whose amplitudes a were varied from 0.1 to 3, and widths at the level of a/2 were equal to triple width of solitons with the same amplitude for a 1) was going on a universal scenario with the generation of nonlinear Airy wave. For the Gardner equation (1) with the positive cubic nonlinearity coefficient there exist two one-parametric families of
Classification of Dark Modified KdV Equation
NASA Astrophysics Data System (ADS)
Xiong, Na; Lou, Sen-Yue; Li, Biao; Chen, Yong
2017-07-01
The dark Korteweg-de Vries (KdV) systems are defined and classified by Kupershmidt sixteen years ago. However, there is no other classifications for other kinds of nonlinear systems. In this paper, a complete scalar classification for dark modified KdV (MKdV) systems is obtained by requiring the existence of higher order differential polynomial symmetries. Different to the nine classes of the dark KdV case, there exist twelve independent classes of the dark MKdV equations. Furthermore, for the every class of dark MKdV system, there is a free parameter. Only for a fixed parameter, the dark MKdV can be related to dark KdV via suitable Miura transformation. The recursion operators of two classes of dark MKdV systems are also given. Supported by the Global Change Research Program of China under Grant No. 2015Cb953904, National Natural Science Foundation of China under Grant Nos. 11675054, 11435005, 11175092, and 11205092 and Shanghai Knowledge Service Platform for Trustworthy Internet of Things (No. ZF1213) and K. C. Wong Magna Fund in Ningbo University
Brazhnyi, V A; Konotop, V V
2005-08-01
The dynamics of vector dark solitons in two-component Bose-Einstein condensates is studied within the framework of coupled one-dimensional nonlinear Schrödinger (NLS) equations. We consider the small-amplitude limit in which the coupled NLS equations are reduced to coupled Korteweg-de Vries (KdV) equations. For a specific choice of the parameters the obtained coupled KdV equations are exactly integrable. We find that there exist two branches of (slow and fast) dark solitons corresponding to the two branches of the sound waves. Slow solitons, corresponding to the lower branch of the acoustic wave, appear to be unstable and transform during the evolution into stable fast solitons (corresponding to the upper branch of the dispersion law). Vector dark solitons of arbitrary depths are studied numerically. It is shown that effectively different parabolic traps, to which the two components are subjected, cause an instability of the solitons, leading to a splitting of their components and subsequent decay. A simple phenomenological theory, describing the oscillations of vector dark solitons in a magnetic trap, is proposed.
1988-01-22
DES MATEIRIAUX A M1E1OIRE DE FORMlE Michel FREIIOND(*) On donne un mod~le thermodynamique macroscopique des mat6rlaux 6 * m6moire de formne utilisant...480. [10] D.J. Korteweg, Archives NGerlandaises, 28, 1901, p. 1-24. [II] Y. Rocard, Thermodynamique , Masson, 1952. [12] J.S. Rowlinson and B. Widom
Some exact solutions to the Lighthill-Whitham-Richards-Payne traffic flow equations
NASA Astrophysics Data System (ADS)
Rowlands, G.; Infeld, E.; Skorupski, A. A.
2013-09-01
We find a class of exact solutions to the Lighthill-Whitham-Richards-Payne (LWRP) traffic flow equations. Using two consecutive Lagrangian transformations, a linearization is achieved. Next, depending on the initial density, we either apply (again two) Lambert functions and obtain exact formulae for the dependence of the car density and velocity on x, t, or else, failing that, the same result in a parametric representation. The calculation always involves two possible factorizations of a consistency condition. Both must be considered. In physical terms, the lineup usually separates into two offshoots at different velocities. Each velocity soon becomes uniform. This outcome in many ways resembles the two soliton solution to the Korteweg-de Vries equation. We check general conservation requirements. Although traffic flow research has developed tremendously since LWRP, this calculation, being exact, may open the door to solving similar problems, such as gas dynamics or water flow in rivers. With this possibility in mind, we outline the procedure in some detail at the end.
NASA Astrophysics Data System (ADS)
Infeld, E.; Rowlands, G.; Skorupski, A. A.
2014-10-01
We find a further class of exact solutions to the Lighthill-Whitham- Richards-Payne (LWRP) traffic flow equations. As before, using two consecutive Lagrangian transformations, a linearization is achieved. Next, depending on the initial density, we either obtain exact formulae for the dependence of the car density and velocity on x,t, or else, failing that, the same result in a parametric representation. The calculation always involves two possible factorizations of a consistency condition. Both must be considered. In physical terms, the lineup usually separates into two offshoots at different velocities. Each velocity soon becomes uniform. This outcome in many ways resembles not only that of Rowlands et al (2013 J. Phys. A: Math. Theor. 46 365202 (part I)) but also the two-soliton solution to the Korteweg-de Vries equation. This paper can be read independently of part I. This explains unavoidable repetitions. Possible uses of both papers in checking numerical codes are indicated. Since LWRP, numerous more elaborate models, including multiple lanes, traffic jams, tollgates, etc, abound in the literature. However, we present an exact solution. These are few and far between, other than found by inverse scattering. The literature for various models, including ours, is given. The methods used here and in part I may be useful in solving other problems, such as shallow water flow.
A New Boussinesq-Type Model for Surface Water Wave Propagation
1998-01-01
velocity-related variable (e.g. depth-averaged velocity, total mass flux, velocity potential at the bottom, etc). Korteweg and deVries (1895) used the same...multiplying each expansion by a coefficient and solving the system of equations resulting from setting the combination of coefficients of the higher...authors have found approximate solutions for the solitary wave, in- cluding the early works of Boussinesq (1871) and Korteweg and deVries (1895
Dust acoustic shock waves in two temperatures charged dusty grains
El-Shewy, E. K.; Abdelwahed, H. G.; Elmessary, M. A.
2011-11-15
The reductive perturbation method has been used to derive the Korteweg-de Vries-Burger equation and modified Korteweg-de Vries-Burger for dust acoustic shock waves in a homogeneous unmagnetized plasma having electrons, singly charged ions, hot and cold dust species with Boltzmann distributions for electrons and ions in the presence of the cold (hot) dust viscosity coefficients. The behavior of the shock waves in the dusty plasma has been investigated.
Koolen, David A; Pfundt, Rolph; Linda, Katrin; Beunders, Gea; Veenstra-Knol, Hermine E; Conta, Jessie H; Fortuna, Ana Maria; Gillessen-Kaesbach, Gabriele; Dugan, Sarah; Halbach, Sara; Abdul-Rahman, Omar A; Winesett, Heather M; Chung, Wendy K; Dalton, Marguerite; Dimova, Petia S; Mattina, Teresa; Prescott, Katrina; Zhang, Hui Z; Saal, Howard M; Hehir-Kwa, Jayne Y; Willemsen, Marjolein H; Ockeloen, Charlotte W; Jongmans, Marjolijn C; Van der Aa, Nathalie; Failla, Pinella; Barone, Concetta; Avola, Emanuela; Brooks, Alice S; Kant, Sarina G; Gerkes, Erica H; Firth, Helen V; Õunap, Katrin; Bird, Lynne M; Masser-Frye, Diane; Friedman, Jennifer R; Sokunbi, Modupe A; Dixit, Abhijit; Splitt, Miranda; Kukolich, Mary K; McGaughran, Julie; Coe, Bradley P; Flórez, Jesús; Nadif Kasri, Nael; Brunner, Han G; Thompson, Elizabeth M; Gecz, Jozef; Romano, Corrado; Eichler, Evan E; de Vries, Bert B A
2016-05-01
The Koolen-de Vries syndrome (KdVS; OMIM #610443), also known as the 17q21.31 microdeletion syndrome, is a clinically heterogeneous disorder characterised by (neonatal) hypotonia, developmental delay, moderate intellectual disability, and characteristic facial dysmorphism. Expressive language development is particularly impaired compared with receptive language or motor skills. Other frequently reported features include social and friendly behaviour, epilepsy, musculoskeletal anomalies, congenital heart defects, urogenital malformations, and ectodermal anomalies. The syndrome is caused by a truncating variant in the KAT8 regulatory NSL complex unit 1 (KANSL1) gene or by a 17q21.31 microdeletion encompassing KANSL1. Herein we describe a novel cohort of 45 individuals with KdVS of whom 33 have a 17q21.31 microdeletion and 12 a single-nucleotide variant (SNV) in KANSL1 (19 males, 26 females; age range 7 months to 50 years). We provide guidance about the potential pitfalls in the laboratory testing and emphasise the challenges of KANSL1 variant calling and DNA copy number analysis in the complex 17q21.31 region. Moreover, we present detailed phenotypic information, including neuropsychological features, that contribute to the broad phenotypic spectrum of the syndrome. Comparison of the phenotype of both the microdeletion and SNV patients does not show differences of clinical importance, stressing that haploinsufficiency of KANSL1 is sufficient to cause the full KdVS phenotype.
NASA Astrophysics Data System (ADS)
Chen, Jinbing
2010-08-01
Each soliton equation in the Korteweg-de Vries (KdV) hierarchy, the 2+1 dimensional breaking soliton equation, and the 2+1 dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation are reduced to two or three Neumann systems on the tangent bundle TSN -1 of the unit sphere SN -1. The Lax-Moser matrix for the Neumann systems of degree N -1 is deduced in view of the Mckean-Trubowitz identity and a bilinear generating function, whose favorite characteristic accounts for the problem of the genus of Riemann surface matching to the number of elliptic variables. From the Lax-Moser matrix, the constrained Hamiltonians in the sense of Dirac-Poisson bracket for all the Neumann systems are written down in a uniform recursively determined by integrals of motion. The involution of integrals of motion and constrained Hamiltonians is completed on TSN -1 by using a Lax equation and their functional independence is displayed over a dense open subset of TSN -1 by a direct calculation, which contribute to the Liouville integrability of a family of Neumann systems in a new systematical way. We also construct the hyperelliptic curve of Riemann surface and the Abel map straightening out the restricted Neumann flows that naturally leads to the Jacobi inversion problem on the Jacobian with the aid of the holomorphic differentials, from which some finite-gap solutions expressed by Riemann theta functions for the 2+1 dimensional breaking soliton equation, the 2+1 dimensional CDGKS equation, the KdV, and the fifth-order KdV equations are presented by means of the Riemann theorem.
NASA Astrophysics Data System (ADS)
Rafat, A.; Rahman, M. M.; Alam, M. S.; Mamun, A. A.
2015-07-01
A precise theoretical investigation has been made on electron-acoustic (EA) Gardner solitons (GSs) and double layers (DLs) in a four-component plasma system consisting of nonextensive hot electrons and positrons, inertial cold electrons, and immobile positive ions. The well-known reductive perturbation method has been used to derive the Korteweg-de Vries (K-dV), modified K-dV (mK-dV), and Gardner equations along with their solitary wave as well as double layer solutions. It has been found that depending on the plasma parameters, the K-dV solitons and GSs are either compressive or rarefactive, whereas the mK-dV solitons are only compressive, and Gardner DLs are only rarefactive. The analytical comparison among the K-dV solitons, mK-dV solitons, and GSs are also investigated. It has been identified that the basic properties of such EA solitons and EA DLs are significantly modified due to the effects of nonextensivity and other plasma parameters related to plasma particle number densities and to temperature of different plasma species. The results of our present investigation can be helpful for understanding the nonlinear electrostatic structures associated with EA waves in various interstellar space plasma environments and cosmological scenarios (viz. quark-gluon plasma, protoneutron stars, stellar polytropes, hadronic matter, dark-matter halos, etc.)
Nonlinear disintegration of sine wave in the framework of the Gardner equation
NASA Astrophysics Data System (ADS)
Kurkin, Andrey; Talipova, Tatiana; Kurkina, Oxana; Rouvinskaya, Ekaterina; Pelinovsky, Efim
2016-04-01
Nonlinear disintegration of sine wave is studied in the framework of the Gardner equation (extended version of the Korteweg - de Vries equation with both quadratic and cubic nonlinear terms). Undular bores appear here as an intermediate stage of wave evolution. Our numerical computations demonstrate the features of undular bore developing for different signs of the cubic nonlinear term. If cubic nonlinear term is negative, and initial wave amplitude is large enough, two undular bores are generated from the two breaking points formed on both crest slopes (within dispersionless Gardner equation). Undular bore consists of one table-top soliton and a group of small soliton-like waves passing through the table-top soliton. If the cubic nonlinear term is positive and again the wave amplitude is large enough, the breaking points appear on crest and trough generating groups of positive and negative solitary-like pulses. It is shown that nonlinear interaction of waves happens according to one of scenarios of two-soliton interaction of "exchange" or "overtake" types with a phase shift. If small-amplitude pulses interact with large-amplitude soliton-like pulses, their speed in average is negative in the case when "free" velocity is positive. Nonlinear interaction leads to the generation of higher harmonics and spectrum width increases with amplitude increase independently of the sign of cubic nonlinear term. The breaking asymptotic k4/3 predicted within the dispersionless Gardner equation emerges during the process of undular bore development. The formation of soliton-like perturbations leads to appearance of several spectral peaks which are downshifting with time.
Modelling Bathymetric Control of Near Coastal Wave Climate. Report 2
1990-04-01
modified form of the Kortweg- deVries equation . 1 Introduction There has recently been a great deal of interest in solving the Boussiresq equations for...shown that model result- based on a truncation of the Fouri, spectrum of the cnoidal 2 wave solution of the Korteweg - deVries (KdV) equation (Flick et...form of the equation is unchanged. edges. Equation (12) may then be solved in the finite domain Explicit results near resonance fo. this case have
Long Internal Waves of Moderate Amplitudes. I. Solitons.
1981-05-01
relea; Dtaribution Unlmjsd k~4.. /" t -. - :,- WNW_ Abstract -The Korteweg - deVries (KdV) equation and the finite-depth equation of Joseph (1977) and...consider two models which are weakly nonlinear and weakly dispersive: the Korteweg - deVries (KdV) equation , fT + 6ffx + f *xx 0 (1) and an equation due to...X). A Galerkin procedure may be used to solve (A.1) approximately. Here (A.1) is replaced by a finite set of equations of the form (L Cv]. *n) (f3
Exact solutions for the (2+1)-dimensional Hirota-Maxwell-Bloch system
NASA Astrophysics Data System (ADS)
Yesmakhanova, Kuralay; Shaikhova, Gaukhar; Bekova, Guldana; Myrzakulov, Ratbay
2017-09-01
In this paper, we consider the (2+1)-dimensional Hirota-Maxwell-Bloch system (HMBS) which with higher order effects usually governs the propagation of ultrashort pulses in nonlinear erbium doped fibers. Integrable condition of such system determined via the associated Lax pair is explicitly constructed. The (2+1)-dimensional HMBS admits reductions such as complex modified Korteweg de Vries-Maxwell-Bloch equations, Hirota system, Schrodinger-Maxwell-Bloch equations, nonlinear Schrodinger equations, complex modified Korteweg de Vries equations. We construct Darboux transformation and provide soliton solutions of the (2+1)-dimensional HMBS by using obtained Darboux transformation.
Higher Order Corrections for Shallow-Water Solitary Waves: Elementary Derivation and Experiments
ERIC Educational Resources Information Center
Halasz, Gabor B.
2009-01-01
We present an elementary method to obtain the equations of the shallow-water solitary waves in different orders of approximation. The first two of these equations are solved to get the shapes and propagation velocities of the corresponding solitary waves. The first-order equation is shown to be equivalent to the Korteweg-de Vries (KdV) equation,…
Higher Order Corrections for Shallow-Water Solitary Waves: Elementary Derivation and Experiments
ERIC Educational Resources Information Center
Halasz, Gabor B.
2009-01-01
We present an elementary method to obtain the equations of the shallow-water solitary waves in different orders of approximation. The first two of these equations are solved to get the shapes and propagation velocities of the corresponding solitary waves. The first-order equation is shown to be equivalent to the Korteweg-de Vries (KdV) equation,…
Small energy traveling waves for the Euler-Korteweg system
NASA Astrophysics Data System (ADS)
Audiard, Corentin
2017-09-01
We investigate the existence and properties of traveling waves for the Euler-Korteweg system with general capillarity and pressure. Our main result is the existence in dimension two of waves with arbitrarily small energies. They are obtained as minimizers of a modified energy with fixed momentum. The proof builds upon various ideas developed for the Gross-Pitaevskii equation (and more generally nonlinear Schrödinger equations with non zero limit at infinity). Even in the Schrödinger case, the fact that we work with the hydrodynamical variables and a general pressure law both brings new difficulties and some simplifications. Independently, in dimension one we prove that the criterion for the linear instability of traveling waves from Benzoni-Gavage (2013 Differ. Integral Equ. 26 439-85) actually implies nonlinear instability.
Refinements to an Optimized Model-Driven Bathymetry Deduction Algorithm
2001-09-01
bathymetric deduction algorithm, we used the Korteweg - deVries (KdV) equation ( Korteweg and deVries 1895) as the wave model. Throughout this study, we will be...technique is explained in an appendix of the manuscript. In the interest of brevity, we simply write the matrix equation to be solved : ηµ ∆+=∆ TTh...the wavelength). Bell (1999) used phase speeds calculated from X-band radar imagery and Equation (1) to infer the bathymetry, with favorable
Numerical Schemes for a Model for Nonlinear Dispersive Waves.
1983-11-01
2604 November 1983 ABSTRACT A description is given of a number of numerical schemes to solve an evolution equation Athat arises when modelling the...travel at constant speed and whose shape is independent of time. One of the models, the Korteweg -de Vries equation , has been studied extensively, both...inital-value problem for the Korteweg -de Vries equation y~~~-2u 0(I) ut + ux + Buu x +fYu inO, Department of Mathematics, University of Chicago, Chicago
Solitons in nucleon-nucleus collisions
Fogaca, D.A.; Navarra, F.S.
2004-12-02
Under certain conditions, the equations of non-relativistic hydrodynamics may provide a Korteweg-de Vries equation (KdV) which gives a soliton solution. We show that this solution and its properties are related to the microscopic features of the nuclear matter equation of state.
Reductions of lattice mKdV to q-PVI
NASA Astrophysics Data System (ADS)
Ormerod, Christopher M.
2012-10-01
This Letter presents a reduction of the lattice modified Korteweg-de Vries equation that gives rise to a q-analogue of the sixth Painlevé equation via a new approach to reductions. This new approach also allows us to give the first ultradiscrete Lax representation of an ultradiscrete analogue of the sixth Painlevé equation.
Quantum positron acoustic waves
Metref, Hassina; Tribeche, Mouloud
2014-12-15
Nonlinear quantum positron-acoustic (QPA) waves are investigated for the first time, within the theoretical framework of the quantum hydrodynamic model. In the small but finite amplitude limit, both deformed Korteweg-de Vries and generalized Korteweg-de Vries equations governing, respectively, the dynamics of QPA solitary waves and double-layers are derived. Moreover, a full finite amplitude analysis is undertaken, and a numerical integration of the obtained highly nonlinear equations is carried out. The results complement our previously published results on this problem.
On dust ion acoustic solitary waves in collisional dusty plasmas with ionization effect
NASA Astrophysics Data System (ADS)
Shalaby, M.; El-Labany, S. K.; El-Shamy, E. F.; Khaled, M. A.
2010-04-01
The propagation of solitary waves in an unmagnetized collisional dusty plasma consisting of a negatively charged dust fluid, positively charged ions, isothermal electrons, and background neutral particles is studied. The ionization, ion loss, ion-neutral, ion-dust, and dust-neutral collisions are considered. Applying a reductive perturbation theory, a damped Korteweg-de Vries (DKdV) equation is derived. On the other hand, at a critical phase velocity, the dynamics of solitary waves is governed by a damped modified Korteweg-de Vries (DMKdV) equation. The nonlinear properties of solitary waves in the two cases are discussed.
On the internal soliton propagation over slope-shelf topography
NASA Astrophysics Data System (ADS)
Sulaiman, Albert
2017-05-01
Dynamics and properties of internal soliton propagation over slope-shelf topography is investigated. We derive a nonlinear wave equation based on the two-layer fluid model, which produce a variable-coefficient perturbed Korteweg-deVries (vP-KdV) equation. A special solution in term of single-soliton will be highlighted.
Shen, Yongqiang; Wang, Lixing; Shao, Renfan; Gong, Tao; Zhu, Chenhui; Yang, Hong; Maclennan, Joseph E; Walba, David M; Clark, Noel A
2013-12-01
In chiral smectic-A (Sm-A) liquid crystals, an applied electric field induces a tilt of the optic axis from the layer normal. When these materials are of the de Vries type, the electroclinic tilt susceptibility is unusually large, with the field-induced director reorientation accompanied by a substantial increase in optical birefringence with essentially no change in the smectic layer spacing. In order to account for the observed electro-optic behavior, we assume that the molecular orientation distribution in the Sm-A has two degrees of freedom: azimuthal orientation and tilt of the molecular long axis from the layer normal, with the tilt confined to a narrow range of angles. We present a generalized Langevin-Debye model of the response of this orientational distribution to applied field that gives a field-induced optic axis tilt, birefringence, and polarization dependence that agrees well with experimental measurements and reproduces the double-peaked polarization current response characteristic of a first-order Sm-A(*)-Sm-C(*) transition. Additionally, we find that the measured field-induced polarization and the Langevin-Debye model predictions can be quantitatively described as pre-transitional behavior near the tricritical point of a recently published generalized 3D XY model of interacting hard rods confined to reorient on a cone in the presence of an applied field.
NASA Astrophysics Data System (ADS)
Reyes, M. A.; Gutiérrez-Ruiz, D.; Mancas, S. C.; Rosu, H. C.
2016-01-01
We present an approach to the bright soliton solution of the nonlinear Schrödinger (NLS) equation from the standpoint of introducing a constant potential term in the equation. We discuss a “nongauge” bright soliton for which both the envelope and the phase depend only on the traveling variable. We also construct a family of generalized NLS equations with solitonic sechp solutions in the traveling variable and find an exact equivalence with other nonlinear equations, such as the Korteveg-de Vries (KdV) and Benjamin-Bona-Mahony (BBM) equations when p = 2.
Research on the Inverse Problem of Scattering
1981-10-01
Levitan equation for the r)ne- dimensional and radial Schroedinger equations., ( b ) provided a vuiri•jtiona1 prine.l pie, and (c) extended inverse techniques...Variational Principle for the Gelfand- Levitan Equation and the Korteweg-deVries Equation (with M . Kanal), J. Math. Phys., 18, 2445 (1977). 3. A...Operators are Identical (with P. B . Abraham and B . DeFaclo), Studies in App. Math. (in press). 9. The Ceifand- Levitan Equation can Give Simple Examples of
Modified ion-acoustic solitary waves in plasmas with field-aligned shear flows
Saleem, H.; Haque, Q.
2015-08-15
The nonlinear dynamics of ion-acoustic waves is investigated in a plasma having field-aligned shear flow. A Korteweg-deVries-type nonlinear equation for a modified ion-acoustic wave is obtained which admits a single pulse soliton solution. The theoretical result has been applied to solar wind plasma at 1 AU for illustration.
Dissipative effects on nonlinear waves in rotating fluids.
NASA Technical Reports Server (NTRS)
Leibovich, S.; Randall, J. D.
1971-01-01
Modifications to the existing inviscid theory of long-wave propagation in rotating fluids are studied. A modification to the Korteweg-deVries equation is found to describe weak dissipation in long waves in a swirling fluid. General features of solutions are discussed, and a solution for the damping of solitary waves is presented.
1989-05-22
1977. 21. Asymptotic Solutions of the Korteweg-deVries Equation, M.J. Ablowitz and H. Segur, Studies in Applied Math., 57, pp. 13-44, 1977. ?2. Exact ... Linearization of a Painleve Transcendent, M.J. Ablowitz and H. Segur, Phys. Rev. Lett., Vol. 38, No. 20, p. 1103, 1977. 23. Solitons and Rational
Modified electron acoustic field and energy applied to observation data
Abdelwahed, H. G. E-mail: hgomaa-eg@mans.edu.eg; El-Shewy, E. K.
2016-08-15
Improved electrostatic acoustic field and energy have been debated in vortex trapped hot electrons and fluid of cold electrons with pressure term plasmas. The perturbed higher-order modified-Korteweg-de Vries equation (PhomKdV) has been worked out. The effect of trapping and electron temperatures on the electro-field and energy properties in auroral plasmas has been inspected.
Brenner, Howard
2014-04-01
"Diffuse interface" theories for single-component fluids—dating back to van der Waals, Korteweg, Cahn-Hilliard, and many others—are currently based upon an ad hoc combination of thermodynamic principles (built largely upon Helmholtz's free-energy potential) and so-called “nonclassical” continuum-thermomechanical principles (built largely upon Newtonian mechanics), with the latter originating with the pioneering work of Dunn and Serrin [Arch. Ration. Mech. Anal. 88, 95 (1985)]. By introducing into the equation governing the transport of energy the notion of an interstitial work-flux contribution, above and beyond the usual Fourier heat-flux contribution, namely, jq = -k∇T, to the energy flux, Dunn and Serrin provided a rational continuum-thermomechanical basis for the presence of Korteweg stresses in the equation governing the transport of linear momentum in compressible fluids. Nevertheless, by their failing to recognize the existence and fundamental need for an independent volume transport equation [Brenner, Physica A 349, 11 (2005)]—especially for the roles played therein by the diffuse volume flux j v and the rate of production of volume πν at a point of the fluid continuum—we argue that diffuse interface theories for fluids stand today as being both ad hoc and incomplete owing to their failure to recognize the need for an independent volume transport equation for the case of compressible fluids. In contrast, we point out that bivelocity hydrodynamics, as it already exists [Brenner, Phys. Rev. E 86, 016307 (2012)], provides a rational, non-ad hoc, and comprehensive theory of diffuse interfaces, not only for single-component fluids, but also for certain classes of crystalline solids [Danielewski and Wierzba, J. Phase Equilib. Diffus. 26, 573 (2005)]. Furthermore, we provide not only what we believe to be the correct constitutive equation for the Korteweg stress in the class of fluids that are constitutively Newtonian in their rheological response
NASA Astrophysics Data System (ADS)
Brenner, Howard
2014-04-01
"Diffuse interface" theories for single-component fluids—dating back to van der Waals, Korteweg, Cahn-Hilliard, and many others—are currently based upon an ad hoc combination of thermodynamic principles (built largely upon Helmholtz's free-energy potential) and so-called "nonclassical" continuum-thermomechanical principles (built largely upon Newtonian mechanics), with the latter originating with the pioneering work of Dunn and Serrin [Arch. Ration. Mech. Anal. 88, 95 (1985)]. By introducing into the equation governing the transport of energy the notion of an interstitial work-flux contribution, above and beyond the usual Fourier heat-flux contribution, namely, jq=-k∇T, to the energy flux, Dunn and Serrin provided a rational continuum-thermomechanical basis for the presence of Korteweg stresses in the equation governing the transport of linear momentum in compressible fluids. Nevertheless, by their failing to recognize the existence and fundamental need for an independent volume transport equation [Brenner, Physica A 349, 11 (2005), 10.1016/j.physa.2004.10.033]—especially for the roles played therein by the diffuse volume flux jv and the rate of production of volume πv at a point of the fluid continuum—we argue that diffuse interface theories for fluids stand today as being both ad hoc and incomplete owing to their failure to recognize the need for an independent volume transport equation for the case of compressible fluids. In contrast, we point out that bivelocity hydrodynamics, as it already exists [Brenner, Phys. Rev. E 86, 016307 (2012), 10.1103/PhysRevE.86.016307], provides a rational, non-ad hoc, and comprehensive theory of diffuse interfaces, not only for single-component fluids, but also for certain classes of crystalline solids [Danielewski and Wierzba, J. Phase Equilib. Diffus. 26, 573 (2005), 10.1007/s11669-005-0002-y]. Furthermore, we provide not only what we believe to be the correct constitutive equation for the Korteweg stress in the
1993-06-03
source frequency is a little larger, so that the next term in the series must be included, the so-called Korteweg - deVries -Burgers equation is obtained...developed by the Bergen group9-1 2 for solving the KZK equation is briefly described and explained. The generalized Westervelt equation , especially its...2.18 reduces to an ordinary integral equation . Even so, the reduced equation is not easy to solve . Li has used the method of multiple scales to
Global Well-Posedness of the Euler-Korteweg System for Small Irrotational Data
NASA Astrophysics Data System (ADS)
Audiard, Corentin; Haspot, Boris
2017-04-01
The Euler-Korteweg equations are a modification of the Euler equations that take into account capillary effects. In the general case they form a quasi-linear system that can be recast as a degenerate Schrödinger type equation. Local well-posedness (in subcritical Sobolev spaces) was obtained by Benzoni-Danchin-Descombes in any space dimension, however, except in some special case (semi-linear with particular pressure) no global well-posedness is known. We prove here that under a natural stability condition on the pressure, global well-posedness holds in dimension {d ≥ 3} for small irrotational initial data. The proof is based on a modified energy estimate, standard dispersive properties if {d ≥ 5}, and a careful study of the structure of quadratic nonlinearities in dimension 3 and 4, involving the method of space time resonances.
Collective Properties of Neural Systems and Their Relation to Other Physical Models
1988-08-05
t and q -- q* are useful transformations in this respect. 4 (d) The existence and uniqueness of solution for the Korteweg-deVries ( KdV ) equation ...Unitversitv. Potsdam. N Y 13676, U, S. A. (Received: I July 1987) Abstract. The Babu-Barouch solution of Berning’s difference equation (or the...Recently, Babu and Barouch [1] obtained an exact analytical solution of Berning’s difference equations in a closed form. This difference equation
Mesoscopi Detailed Balance Algorithms for Quantum and Classical Turbulence
2013-02-01
the one-dimensional Magnetohydrodynamics-Burgers equations, KdV and nonlinear Schrodinger equations. Generalizing to three dimensions, quantum...Solitons We have investigated quantum unitary algorithms for both the 1D Korteweg-de-Vries and the Nonlinear Schrodinger equations [G. Vahala, J...Yepez and L. Vahala, Phys. Lett. A310, 187- 196 (2003)]. In particular to recover the 1D nonlinear Schrodinger equation for bright solitons
Vortex Formation and Particle Transport in a Cross-Field Plasma Sheath.
1988-03-20
appearance of a given kind of nonlinear structure from arbitrary imtial conditions (as can be done, say, in the case of the Korteweg - deVries equation ...rd. _ -. surface charge, which in turn is a boundary source in Poisson’s equation . In solving Poisson’s equation , the program also automatically...coefficients are given, respectively, in Eqs.(89) and (76), and then solve the diffusion equation (72), with appropriate boundary conditions (n(x = 0
1992-09-01
Waves. Wiley, New York. Miles, J. W. 1979. "On the Korteweg - deVries Equation for a Gradually Varying Channel," J.M Vol 91, pp 181-190. 1980. "Solitary... equations , are difficult to solve . One popular 3 approach has been to systematically simplify the three-dimensional equations and their boundary conditions...from three dimensions to two. This theory yields governing equations for the flow, which are solved numerically in a more efficient manner than those
1994-01-03
August, 1991. Thesis - "Applications of the Inverse Spectral Transform to a Korteweg - DeVries Equation with a Kuramoto-Sivashinsky-Type Perturbation... equations , the mathematical theory of nematic optics involves strong coupling between the electromagnetic and nematic director (molecular orientation... equations for the electric field E coupled to a nonlinear parabolic equation for the director n, a field of unit vectors which describes the local molecular
Solving Differential Equations in R: Package deSolve
In this paper we present the R package deSolve to solve initial value problems (IVP) written as ordinary differential equations (ODE), differential algebraic equations (DAE) of index 0 or 1 and partial differential equations (PDE), the latter solved using the method of lines appr...
Solving Differential Equations in R: Package deSolve
In this paper we present the R package deSolve to solve initial value problems (IVP) written as ordinary differential equations (ODE), differential algebraic equations (DAE) of index 0 or 1 and partial differential equations (PDE), the latter solved using the method of lines appr...
Nonlinear waves in a viscous fluid contained in a viscoelastic tube
NASA Astrophysics Data System (ADS)
Demiray, H.
In the present work the propagation of weakly nonlinear waves in a prestressed viscoelastic thin tube filled with a viscous fluid is studied. Using the reductive perturbation technique in analyzing the nonlinear equations of a viscoelastic tube and the approximate equations of a viscous fluid, the propagation of weakly nonlinear waves in the longwave approximation is studied. Depending on the order of viscous effects, various evolution equations like, Burgers', Korteweg-de Vries, Korteweg-de Vries-Burgers' equations and their perturbed forms are obtained. Travelling wave type of solutions to some of these evolution equations are sought. Finally, utilizing the finite difference scheme, a numerical solution is presentede for the perturbed KdVB equation and the result is discussed.
2006-06-01
sech2 wave form is used because the amplitude and horizontal displacement are solutions of the Korteweg de Vries ( KdV ) non linear wave equation which...a solution to the KDV wave equation . After making the frozen field approximation, the soliton can be represented by the following mathematical...scattering. 3. The Gaussian Soliton As discussed, the sech2 form of a soliton is chosen because it is an exact solution to the KDV wave equation . For
Theory and Modeling of Internal Wave Generation in Straits
2012-09-30
2.3 days in the SCS). This work is being finalized for publication. 7 Figure 4. Numerical solutions of the rotating KP equation (2) for an... solutions of these models and solutions of the full Navier-Stokes equations . The theoretical models require some simplifications that, depending on the...fully hydrostatic dynamics. The models used include those in the Korteweg-de Vries ( KdV ) family of equations modified to include higher-order
A new version of the generalized F-expansion method and its applications
NASA Astrophysics Data System (ADS)
Pandir, Yusuf; Turhan, Nail
2017-01-01
In this study, a new version of the generalized F-expansion method is suggested to search exact solutions of nonlinear partial differential equations. We find many new and interesting results for Korteweg-de Vries(KdV) equation by use of the proposed method. The solutions acquired from the proposed method are single and combined non-degenerate Jacobi elliptic function solutions. The new method allows a more systematic, easiness use of the solution process of nonlinear equations.
Nonlinear Problems in Fluid Dynamics and Inverse Scattering
1993-05-31
We have demonstrated that a certain class of multidimensional extensions of the well- known Korteweg - deVries equations , often referred to as higher...ion and multiplication in the wavelets bases. Several additional algorithms relevant to solving the nonlinear equations and capturing the...algorithms for solving n1ow linear equations grew into a sizable effort. I work now with two graduate students. .laies IKeiser and Robert Cramer. With
Wave Turbulence and Soliton Dynamics
1992-04-30
Schrodinger equation . one is a domain wall between different types of vibration, the other is a kink in the phase of vibration. The kink has also been...observational data. The domain walls and noncutoff kinks are new localized structures, and may lead to new generic equations at the level of the NLS, Korteweg-de...Vries, sine-Gordon, and Toda equations . To our knowledge there are no reported observations or theory of vibratory kinks and domain walls. A
An Analytical Model of Periodic Waves in Shallow Water--Summary.
1984-01-01
generalizes the Korteweg-deVries ( KdV ) equation, ut + 6uux U O. (2) Physically, the derivation of (1) is very similar to that of (2), except that the...derivation). 2 .................................................................... .... The KP equation also generalizes the KdV equation in the sense that...usual cnoidal wave solution that is familiar from KdV theory 3 * *. ** * .",. % % X V .. ~*J* *J** ~ . .~ .* . . . .. ~ * ... " "_ (e.g., see Sarpkaya
Phase-modulated solitary waves controlled by a boundary condition at the bottom.
Mukherjee, Abhik; Janaki, M S
2014-06-01
A forced Korteweg-de Vries (KdV) equation is derived to describe weakly nonlinear, shallow-water surface wave propagation over nontrivial bottom boundary condition. We show that different functional forms of bottom boundary conditions self-consistently produce different forced KdV equations as the evolution equations for the free surface. Solitary wave solutions have been analytically obtained where phase gets modulated controlled by bottom boundary condition, whereas amplitude remains constant.
Formation of wave packets in the Ostrovsky equation for both normal and anomalous dispersion
Grimshaw, Roger; Stepanyants, Yury; Alias, Azwani
2016-01-01
It is well known that the Ostrovsky equation with normal dispersion does not support steady solitary waves. An initial Korteweg–de Vries solitary wave decays adiabatically through the radiation of long waves and is eventually replaced by an envelope solitary wave whose carrier wave and envelope move with different velocities (phase and group velocities correspondingly). Here, we examine the same initial condition for the Ostrovsky equation with anomalous dispersion, when the wave frequency increases with wavenumber in the limit of very short waves. The essential difference is that now there exists a steady solitary wave solution (Ostrovsky soliton), which in the small-amplitude limit can be described asymptotically through the solitary wave solution of a nonlinear Schrödinger equation, based at that wavenumber where the phase and group velocities coincide. Long-time numerical simulations show that the emergence of this steady envelope solitary wave is a very robust feature. The initial Korteweg–de Vries solitary wave transforms rapidly to this envelope solitary wave in a seemingly non-adiabatic manner. The amplitude of the Ostrovsky soliton strongly correlates with the initial Korteweg–de Vries solitary wave. PMID:26997887
Transformed Flux-Form Semi-Lagrangian Scheme
2008-01-01
5) which is the Korteweg-de Vries ( KDV ) equation with the exact solution , ( ) ( )2 2, sechs t A...term. Evidently equation (7) has the analytical solution (6), and therefore it can be used to verify the stability of the numerical schemes since...in ocean models since any diffusion and dispersion in the numerical solution of the Rossby soliton are computational errors. Interested readers are
Space-Time Transformation in Flux-form Semi-Lagrangian Schemes
2010-02-01
f f1 5366 0 0987651 2= = (5) which is the Korteweg-de Vries ( KDV ) equation with the exact solution , h( ) ( ), secs t A B s B t2 2h n...dispersion into the approximate solution . The numerical diffusion and dispersion are aliens to the process that is being modeled (Chu and Fan 1998, 1999...some artificial viscosity is introduced. Hence, less the numerical diffusion and dispersion errors equates to better model performance. Many
Integrated Modeling and Analysis of Physical Oceanographic and Acoustic Processes
2011-09-01
deVries type wave evolution equations and 2D NHP numerical models. 3. Improved 4D deterministic and stochastic acoustic modeling. Improvements to time...Specifically, an analog of the rotation-neglecting Taylor-Goldstein equation was solved , after making reasonable simplifying assumptions. The...positions and sizes than the full NHP model (task 1), but may sacrifice detail and accuracy. Candidate models include those based on Korteweg
Dressed soliton in quantum dusty pair-ion plasma
Chatterjee, Prasanta; Muniandy, S. V.; Wong, C. S.; Roy, Kaushik
2009-11-15
Nonlinear propagation of a quantum ion-acoustic dressed soliton is studied in a dusty pair-ion plasma. The Korteweg-de Vries (KdV) equation is derived using reductive perturbation technique. A higher order inhomogeneous differential equation is obtained for the higher order correction. The expression for a dressed soliton is calculated using a renormalization method. The expressions for higher order correction are determined using a series solution technique developed by Chatterjee et al. [Phys. Plasmas 16, 072102 (2009)].
Modulations of perturbed KdV wavetrains
Forest, M.G.; Mclaughlin, D.W.
1984-04-01
The modulations of N-phase Korteweg-de Vries (KdV) wavetrains in the presence of external perturbations is investigated. An invariant representation of these modulation equations in terms of differentials on a Riemann surface is derived from averaged perturbed conservation laws. In particular, the explicit dependence of the representation on the external perturbation is obtained. This invariant representation is used to place the equation in a Riemann diagonal form, whose dependence on the external perturbation is explicitly computed. 15 references.
Choudhury, Sourav; Das, Tushar Kanti; Chatterjee, Prasanta; Ghorui, Malay Kr.
2016-06-15
The influence of exchange-correlation potential, quantum Bohm term, and degenerate pressure on the nature of solitary waves in a quantum semiconductor plasma is investigated. It is found that an amplitude and a width of the solitary waves change with variation of different parameters for different semiconductors. A deformed Korteweg-de Vries equation is obtained for propagation of nonlinear waves in a quantum semiconductor plasma, and the effects of different plasma parameters on the solution of the equation are also presented.
On a plasma having nonextensive electrons and positrons: Rogue and solitary wave propagation
El-Awady, E. I.; Moslem, W. M.
2011-08-15
Generation of nonlinear ion-acoustic waves in a plasma having nonextensive electrons and positrons has been studied. Two wave modes existing in such plasma are considered, namely solitary and rogue waves. The reductive perturbation method is used to obtain a Korteweg-de Vries equation describing the system. The latter admits solitary wave pulses, while the dynamics of the modulationally unstable wave packets described by the Korteweg-de Vries equation gives rise to the formation of rogue excitation that is described by a nonlinear Schroedinger equation. The dependence of both solitary and rogue waves profiles on the nonextensive parameter, positron-to-ion concentration ratio, electron-to-positron temperature ratio, and ion-to-electron temperature ratio are investigated numerically. The results from this work are expected to contribute to the in-depth understanding of the nonlinear excitations that may appear in nonextensive astrophysical plasma environments, such as galactic clusters, interstellar medium, etc.
Mushtaq, A.; Saeed, R.; Haque, Q.
2011-04-15
Linear and nonlinear coupled electrostatic drift and ion acoustic waves are studied in inhomogeneous, collisional pair ion-electron plasma. The Korteweg-de Vries-Burgers (KdVB) equation for a medium where both dispersion and dissipation are present is derived. An attempt is made to obtain exact solution of KdVB equation by using modified tanh-coth method for arbitrary velocity of nonlinear drift wave. Another exact solution for KdVB is obtained, which gives a structure of shock wave. Korteweg-de Vries (KdV) and Burgers equations are derived in limiting cases with solitary and monotonic shock solutions, respectively. Effects of species density, magnetic field, obliqueness, and the acoustic to drift velocity ratio on the solitary and shock solutions are investigated. The results discussed are useful in understanding of low frequency electrostatic waves at laboratory pair ion plasmas.
Formation of quasiparallel Alfven solitons
NASA Technical Reports Server (NTRS)
Hamilton, R. L.; Kennel, C. F.; Mjolhus, E.
1992-01-01
The formation of quasi-parallel Alfven solitons is investigated through the inverse scattering transformation (IST) for the derivative nonlinear Schroedinger (DNLS) equation. The DNLS has a rich complement of soliton solutions consisting of a two-parameter soliton family and a one-parameter bright/dark soliton family. In this paper, the physical roles and origins of these soliton families are inferred through an analytic study of the scattering data generated by the IST for a set of initial profiles. The DNLS equation has as limiting forms the nonlinear Schroedinger (NLS), Korteweg-de-Vries (KdV) and modified Korteweg-de-Vries (MKdV) equations. Each of these limits is briefly reviewed in the physical context of quasi-parallel Alfven waves. The existence of these limiting forms serves as a natural framework for discussing the formation of Alfven solitons.
Internal solitary waves propagating through variable background hydrology and currents
NASA Astrophysics Data System (ADS)
Liu, Z.; Grimshaw, R.; Johnson, E.
2017-08-01
Large-amplitude, horizontally-propagating internal wave trains are commonly observed in the coastal ocean, fjords and straits. They are long nonlinear waves and hence can be modelled by equations of the Korteweg-de Vries type. However, typically they propagate through regions of variable background hydrology and currents, and over variable bottom topography. Hence a variable-coefficient Korteweg-de Vries equation is needed to model these waves. Although this equation is now well-known and heavily used, a term representing non-conservative effects, arising from dissipative or forcing terms in the underlying basic state, has usually been omitted. In particular this term arises when the hydrology varies in the horizontal direction. Our purpose in this paper is to examine the possible significance of this term. This is achieved through analysis and numerical simulations, using both a two-layer fluid model and a re-examination of previous studies of some specific ocean cases.
Automating prescription map building for VRI systems using plant feedback
USDA-ARS?s Scientific Manuscript database
Prescription maps for commercial variable rate irrigation (VRI) equipment direct the irrigation rates for each sprinkler zone on a sprinkler lateral as the lateral moves across the field. Typically, these maps are manually uploaded at the beginning of the irrigation season; and the maps are based on...
Dust ion-acoustic solitary waves in a dusty plasma with nonextensive electrons.
Bacha, Mustapha; Tribeche, Mouloud; Shukla, Padma Kant
2012-05-01
The dust-modified ion-acoustic waves of Shukla and Silin are revisited within the theoretical framework of the Tsallis statistical mechanics. Nonextensivity may originate from correlation or long-range plasma interactions. Interestingly, we find that owing to electron nonextensivity, dust ion-acoustic (DIA) solitary waves may exhibit either compression or rarefaction. Our analysis is then extended to include self-consistent dust charge fluctuation. In this connection, the correct nonextensive electron charging current is rederived. The Korteweg-de Vries equation, as well as the Korteweg-de Vries-Burgers equation, is obtained, making use of the reductive perturbation method. The DIA waves are then analyzed for parameters corresponding to space dusty plasma situations.
Dust ion-acoustic solitary waves in a dusty plasma with nonextensive electrons
NASA Astrophysics Data System (ADS)
Bacha, Mustapha; Tribeche, Mouloud; Shukla, Padma Kant
2012-05-01
The dust-modified ion-acoustic waves of Shukla and Silin are revisited within the theoretical framework of the Tsallis statistical mechanics. Nonextensivity may originate from correlation or long-range plasma interactions. Interestingly, we find that owing to electron nonextensivity, dust ion-acoustic (DIA) solitary waves may exhibit either compression or rarefaction. Our analysis is then extended to include self-consistent dust charge fluctuation. In this connection, the correct nonextensive electron charging current is rederived. The Korteweg-de Vries equation, as well as the Korteweg-de Vries-Burgers equation, is obtained, making use of the reductive perturbation method. The DIA waves are then analyzed for parameters corresponding to space dusty plasma situations.
Dust-Ion-Acoustic Solitary and Shock Structures in Multi-Ion Plasmas with Super-Thermal Electrons
NASA Astrophysics Data System (ADS)
Haider, Md. Masum; Nahar, Aynoon
2017-07-01
The propagation of dust-ion-acoustic (DIA) solitary and shock waves in multi-ion (MI) unmagnetised and magnetised plasmas have been studied theoretically. The plasma system contains positively and negatively charged inertial ions, opposite polarity dusts, and high energetic super-thermal electrons. The fluid equations in the system are reduced to a Korteweg-de Vries (K-dV) and Korteweg-de Vries Burger (K-dVB) equations in the limit of small amplitude perturbation. The effect of super-thermal electrons, the opposite polarity of ions, and dusts in the solitary and shock waves are presented graphically and numerically. Present investigations will help to astrophysical and laboratory plasmas.
Robustness of de Saint Venant equations for simulating unsteady flows
Baltzer, Robert A.; Schaffranek, Raymond W.; Lai, Chintu; ,
1995-01-01
Long-wave motion in open channels can be expressed mathematically by the one-dimensional de Saint Venant equations describing conservation of fluid mass and momentum. Numerical simulation models, based on either depth/velocity or water-level/discharge dependent-variable formulations of these equations, are typically used to simulate unsteady open-channel flow. However, the implications and significance of selecting either dependent-variable form - on model development, discretization and numerical solution processes, and ultimately on the range-of-application and simulation utility of resulting models - are not well known. Results obtained from a set of numerical experiments employing two models - one based on depth/velocity and the other on water-level/discharge equation formulations - reveal the sensitivity of the two equation sets to various channel properties and dynamic flow conditions. In particular, the effects of channel gradient, channel width-to-depth ratio, flow-resistance coefficient, and flow unsteadiness are analyzed and discussed.
Peakompactons: Peaked compact nonlinear waves
NASA Astrophysics Data System (ADS)
Christov, Ivan C.; Kress, Tyler; Saxena, Avadh
2017-04-01
This paper is meant as an accessible introduction to/tutorial on the analytical construction and numerical simulation of a class of nonstandard solitary waves termed peakompactons. These peaked compactly supported waves arise as solutions to nonlinear evolution equations from a hierarchy of nonlinearly dispersive Korteweg-de Vries-type models. Peakompactons, like the now-well-known compactons and unlike the soliton solutions of the Korteweg-de Vries equation, have finite support, i.e., they are of finite wavelength. However, unlike compactons, peakompactons are also peaked, i.e., a higher spatial derivative suffers a jump discontinuity at the wave’s crest. Here, we construct such solutions exactly by reducing the governing partial differential equation to a nonlinear ordinary differential equation and employing a phase-plane analysis. A simple, but reliable, finite-difference scheme is also designed and tested for the simulation of collisions of peakompactons. In addition to the peakompacton class of solutions, the general physical features of the so-called K#(n,m) hierarchy of nonlinearly dispersive Korteweg-de Vries-type models are discussed as well.
Fast Integration of One-Dimensional Boundary Value Problems
NASA Astrophysics Data System (ADS)
Campos, Rafael G.; Ruiz, Rafael García
2013-11-01
Two-point nonlinear boundary value problems (BVPs) in both unbounded and bounded domains are solved in this paper using fast numerical antiderivatives and derivatives of functions of L2(-∞, ∞). This differintegral scheme uses a new algorithm to compute the Fourier transform. As examples we solve a fourth-order two-point boundary value problem (BVP) and compute the shape of the soliton solutions of a one-dimensional generalized Korteweg-de Vries (KdV) equation.
Scherbakov, A V; van Capel, P J S; Akimov, A V; Dijkhuis, J I; Yakovlev, D R; Berstermann, T; Bayer, M
2007-08-03
Acoustic solitons formed during the propagation of a picosecond strain pulse in a GaAs crystal with a ZnSe/ZnMgSSe quantum well on top lead to exciton resonance energy shifts of up to 10 meV, and ultrafast frequency modulation, i.e., chirping, of the exciton transition. The effects are well described by a theoretical analysis based on the Korteweg-de Vries equation and accounting for the properties of the excitons in the quantum well.
Recurrence of initial state of nonlinear ion waves
Abe, K.; Satofuka, N.
1981-06-01
By solving the Korteweg--deVries equation in a wide range of the ratio between the nonlinearity and the dispersion, the recurrence of the initial state of the ion wave is examined. The recurrence is assured of taking place only when the dispersion of the initial ion wave predominates over the nonlinearity. If the initial wave has strong nonlinearity compared with the dispersion, the recurrence is indistinct, and the initial monochromatic wave evolves to a turbulent state.
NASA Astrophysics Data System (ADS)
Shubina, Maria
2016-09-01
In this paper, we investigate the one-dimensional parabolic-parabolic Patlak-Keller-Segel model of chemotaxis. For the case when the diffusion coefficient of chemical substance is equal to two, in terms of travelling wave variables the reduced system appears integrable and allows the analytical solution. We obtain the exact soliton solutions, one of which is exactly the one-soliton solution of the Korteweg-de Vries equation.
Few-cycle optical solitons in linearly coupled waveguides
NASA Astrophysics Data System (ADS)
Terniche, Said; Leblond, Hervé; Mihalache, Dumitru; Kellou, Abdelhamid
2016-12-01
We consider soliton propagation in two parallel optical waveguides, in the presence of a linear nondispersive coupling and in the few-cycle regime. The numerical analysis is based on a set of two coupled modified Korteweg-de Vries equations. The evidenced few-cycle vector solitons are optical breathers. In addition to the usual breathing due to carrier-envelope velocity mismatch, we observe, and describe in detail, spatial oscillations of soliton's amplitude and energy.
On certain families of rational functions arising in dynamics
NASA Technical Reports Server (NTRS)
Byrnes, C. I.
1979-01-01
It is noted that linear systems, depending on parameters, can occur in diverse situations including families of rational solutions to the Korteweg-de Vries equation or to the finite Toda lattice. The inverse scattering method used by Moser (1975) to obtain canonical coordinates for the finite homogeneous Toda lattice can be used for the synthesis of RC networks. It is concluded that the multivariable RC setting is ideal for the analysis of the periodic Toda lattice.
Cylindrical and spherical ion acoustic waves in a plasma with nonthermal electrons and warm ions
Sahu, Biswajit; Roychoudhury, Rajkumar
2005-05-15
Using the reductive perturbation technique, nonlinear cylindrical and spherical Korteweg-de Vries (KdV) and modified KdV equations are derived for ion acoustic waves in an unmagnetized plasma consisting of warm adiabatic ions and nonthermal electrons. The effects of nonthermally distributed electrons on cylindrical and spherical ion acoustic waves are investigated. It is found that the nonthermality has a very significant effect on the nature of ion acoustic waves.
Evolution of solitons over a randomly rough seabed.
Mei, Chiang C; Li, Yile
2004-01-01
For long waves propagating over a randomly uneven seabed, we derive a modified Korteweg-de Vries (KdV) equation including new terms representing the effects of disorder on amplitude attenuation and wave phase. Analytical and numerical results are described for the evolution of a soliton entering a semi-infinite region of disorder, and the fission of new solitons after passing over a finite region of disorder.
Nonplanar ion-acoustic solitary waves with superthermal electrons in warm plasma
Eslami, Parvin; Mottaghizadeh, Marzieh; Pakzad, Hamid Reza
2011-07-15
In this paper, we consider an unmagnetized plasma consisting of warm adiabatic ions, superthermal electrons, and thermal positrons. Nonlinear cylindrical and spherical modified Korteweg-de Vries (KdV) equations are derived for ion acoustic waves by using reductive perturbation technique. It is observed that an increasing positron concentration decreases the amplitude of the waves. Furthermore, the effects of the superthermal parameter (k) on the ion acoustic waves are found.
Magnetoacoustic solitons in quantum plasma
Hussain, S.; Mahmood, S.
2011-08-15
Nonlinear magnetoacoustic waves in collisionless homogenous, magnetized quantum plasma is studied. Two fluid quantum magneto-hydrodynamic model (QMHD) is employed and reductive perturbation method is used to derive Korteweg de Vries (KdV) equation for magnetoacoustic waves. The effects of plasma density and magnetic field intensity are investigated on magnetoacoustic solitary structures in quantum plasma. The numerical results are also presented, which are applicable to explain some aspects of the propagation of nonlinear magnetoacosutic wave in dense astrophysical plasma situations.
Effects of Landau damping on finite amplitude low-frequency nonlinear waves in a dusty plasma
NASA Astrophysics Data System (ADS)
Sikdar, Arnab; Khan, Manoranjan
2017-06-01
The effect of linear ion Landau damping on weakly nonlinear as well as weakly dispersive low-frequency waves in a dusty plasma is investigated. The standard perturbative approach leads to the Korteweg-de Vries (KdV) equation with a linear Landau damping term for the dynamics of the low-frequency nonlinear wave. Landau damping causes the wave amplitude to decay with time and the dust charge variation enhances the damping rate.
NASA Astrophysics Data System (ADS)
Kaur, Harvinder; Gill, Tarsem Singh; Bala, Parveen
2017-08-01
In the present investigation, ion-acoustic double layers in an inhomogeneous plasma consisting of Maxwellian and non-thermal distributions of electrons are studied. We have derived a modified Korteweg-de Vries (mKdV) equation for ion-acoustic double layers propagating in a collisionless inhomogeneous plasma. It is observed that the non-thermal parameters affect the amplitude and width of the double layer which further depend on the density.
On certain families of rational functions arising in dynamics
NASA Technical Reports Server (NTRS)
Byrnes, C. I.
1979-01-01
It is noted that linear systems, depending on parameters, can occur in diverse situations including families of rational solutions to the Korteweg-de Vries equation or to the finite Toda lattice. The inverse scattering method used by Moser (1975) to obtain canonical coordinates for the finite homogeneous Toda lattice can be used for the synthesis of RC networks. It is concluded that the multivariable RC setting is ideal for the analysis of the periodic Toda lattice.
Numerical schemes for a model for nonlinear dispersive waves
NASA Technical Reports Server (NTRS)
Bona, J. L.; Pritchard, W. G.; Scott, L. R.
1985-01-01
A description is given of a number of numerical schemes to solve an evolution equation (Korteweg-deVries) that arises when modelling the propagation of water waves in a channel. The discussion also includes the results of numerical experiments made with each of the schemes. It is suggested, on the basis of these experiments, that one of the schemes may have (discrete) solitary-wave solutions.
Mayout, Saliha; Tribeche, Mouloud; Sahu, Biswajit
2015-12-15
A theoretical study on the nonlinear propagation of nonplanar (cylindrical and spherical) dust ion-acoustic solitary waves (DIASW) is carried out in a dusty plasma, whose constituents are inertial ions, superthermal electrons, and charge fluctuating stationary dust particles. Using the reductive perturbation theory, a modified Korteweg-de Vries equation is derived. It is shown that the propagation characteristics of the cylindrical and spherical DIA solitary waves significantly differ from those of their one-dimensional counterpart.
Soliton Perturbations of the Charged Dislocation Core in a Semiconductor Crystal
NASA Astrophysics Data System (ADS)
Gestrin, S. G.; Shchukina, E. V.
2017-04-01
We consider the wave processes in a hole gas inside an electric field created by the charge distribution of donors and acceptors near a negatively charged dislocation in a semiconductor crystal of n-type. It is shown that the solution of the Korteweg-de Vries equation describes solitary waves propagating along the axis of the Read's cylinder. The soliton velocity is estimated for the values of physical parameters characterizing the semiconductor crystal and the region near the dislocation.
Orbital stability and asymptotic stability of mKdV breather-type soliton solutions
NASA Astrophysics Data System (ADS)
Wang, Juan; Tian, Lixin; Zhang, Yingnan
2017-04-01
In this paper, we study the stability of modified Korteweg-de Vries equation breather. By using variable separation method, we obtain the exact breather-type soliton solutions. Moreover, this kind of solutions are globally stable in H2 topology, and we describe a simple, mathematical proof of the orbital stability and asymptotic stability of breather-type soliton solutions under a class of small perturbation.
NASA Astrophysics Data System (ADS)
Peng, Guanghan; Qing, Li
2016-06-01
In this paper, a new car-following model is proposed by considering the drivers’ aggressive characteristics. The stable condition and the modified Korteweg-de Vries (mKdV) equation are obtained by the linear stability analysis and nonlinear analysis, which show that the drivers’ aggressive characteristics can improve the stability of traffic flow. Furthermore, the numerical results show that the drivers’ aggressive characteristics increase the stable region of traffic flow and can reproduce the evolution and propagation of small perturbation.
Identification and determination of solitary wave structures in nonlinear wave propagation
Newman, W.I.; Campbell, D.K.; Hyman, J.M.
1991-01-01
Nonlinear wave phenomena are characterized by the appearance of solitary wave coherent structures'' traveling at speeds determined by their amplitudes and morphologies. Assuming that these structures are briefly noninteracting, we propose a method for the identification of the number of independent features and their respective speeds. Using data generated from an exact two-soliton solution to the Korteweg-de-Vries equation, we test the method and discuss its strengths and limitations. 41 refs., 2 figs.
Uniform strongly interacting soliton gas in the frame of the Nonlinear Schrodinger Equation
NASA Astrophysics Data System (ADS)
Gelash, Andrey; Agafontsev, Dmitry
2017-04-01
support of the Russian Science Foundation (Grand No. 14-22-00174) [1] D. Dutykh, E. Pelinovsky, Numerical simulation of a solitonic gas in kdv and kdv-bbm equations, Physics Letters A 378 (42) (2014) 3102-3110. [2] E. Shurgalina, E. Pelinovsky, Nonlinear dynamics of a soliton gas: Modified korteweg-de vries equation framework, Physics Letters A 380 (24) (2016) 2049-2053. [3] E. N. Pelinovsky, E. Shurgalina, A. Sergeeva, T. G. Talipova, G. El, R. H. Grimshaw, Two-soliton interaction as an elementary act of soliton turbulence in integrable systems, Physics Letters A 377 (3) (2013) 272-275 [4] J. Soto-Crespo, N. Devine, N. Akhmediev, Integrable turbulence and rogue waves: Breathers or solitons?, Physical review letters 116 (10) (2016) 103901. [5] D. S. Agafontsev, V. E. Zakharov, Integrable turbulence and formation of rogue waves, Nonlinearity 28 (8) (2015) 2791. [6] V. E. Zakharov, A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP 34 (1) (1972) 62. [7] V. Zakharov, A. Mikhailov, Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method, Sov. Phys.-JETP (Engl. Transl.) 47 (6) (1978). [8] A. A. Gelash, V. E. Zakharov, Superregular solitonic solutions: a novel scenario for the nonlinear stage of modulation instability, Nonlinearity 27 (4) (2014) R1.
A dimension-breaking phenomenon in the theory of steady gravity-capillary water waves.
Groves, M D; Haragus, M; Sun, S M
2002-10-15
The existence of a line solitary-wave solution to the water-wave problem with strong surface-tension effects was predicted on the basis of a model equation in the celebrated 1895 paper by D. J. Korteweg and G. de Vries and rigorously confirmed a century later by C. J. Amick and K. Kirchgässner in 1989. A model equation derived by B. B. Kadomtsev and V. I. Petviashvili in 1970 suggests that the Korteweg-de Vries line solitary wave belongs to a family of periodically modulated solitary waves which have a solitary-wave profile in the direction of motion and are periodic in the transverse direction. This prediction is rigorously confirmed for the full water-wave problem in the present paper. It is shown that the Korteweg-de Vries solitary wave undergoes a dimension-breaking bifurcation that generates a family of periodically modulated solitary waves. The term dimension-breaking phenomenon describes the spontaneous emergence of a spatially inhomogeneous solution of a partial differential equation from a solution which is homogeneous in one or more spatial dimensions.
A novel lattice traffic flow model on a curved road
NASA Astrophysics Data System (ADS)
Cao, Jin-Liang; Shi, Zhon-Ke
2015-03-01
Due to the existence of curved roads in real traffic situation, a novel lattice traffic flow model on a curved road is proposed by taking the effect of friction coefficient and radius into account. The stability condition is obtained by using linear stability theory. The result shows that the traffic flow becomes stable with the decrease of friction coefficient and radius of the curved road. Using nonlinear analysis method, the Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equation are derived to describe soliton waves and the kink-antikink waves in the meta-stable region and unstable region, respectively. Numerical simulations are carried out and the results are consistent with the theoretical results.
Capillary solitons on a levitated medium
NASA Astrophysics Data System (ADS)
Perrard, S.; Deike, L.; Duchêne, C.; Pham, C.-T.
2015-07-01
A water cylinder deposited on a heated channel levitates on its own generated vapor film owing to the Leidenfrost effect. This experimental setup permits the study of the one-dimensional propagation of surface waves in a free-to-move liquid system. We report the observation of gravity-capillary waves under a dramatic reduction of gravity (up to a factor 30), leading to capillary waves at the centimeter scale. The generated nonlinear structures propagate without deformation and undergo mutual collisions and reflections at the boundaries of the domain. They are identified as Korteweg-de Vries solitons with negative amplitude and subsonic velocity. The typical width and amplitude-dependent velocities are in excellent agreement with theoretical predictions based on a generalized Korteweg-de Vries equation adapted to any substrate geometry. When multiple solitons are present, they interact and form a soliton turbulencelike spectrum.
An extended optimal velocity difference model in a cooperative driving system
NASA Astrophysics Data System (ADS)
Cao, Jinliang; Shi, Zhongke; Zhou, Jie
2015-10-01
An extended optimal velocity (OV) difference model is proposed in a cooperative driving system by considering multiple OV differences. The stability condition of the proposed model is obtained by applying the linear stability theory. The results show that the increase in number of cars that precede and their OV differences lead to the more stable traffic flow. The Burgers, Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations are derived to describe the density waves in the stable, metastable and unstable regions, respectively. To verify these theoretical results, the numerical simulation is carried out. The theoretical and numerical results show that the stabilization of traffic flow is enhanced by considering multiple OV differences. The traffic jams can be suppressed by taking more information of cars ahead.
Capillary solitons on a levitated medium.
Perrard, S; Deike, L; Duchêne, C; Pham, C-T
2015-07-01
A water cylinder deposited on a heated channel levitates on its own generated vapor film owing to the Leidenfrost effect. This experimental setup permits the study of the one-dimensional propagation of surface waves in a free-to-move liquid system. We report the observation of gravity-capillary waves under a dramatic reduction of gravity (up to a factor 30), leading to capillary waves at the centimeter scale. The generated nonlinear structures propagate without deformation and undergo mutual collisions and reflections at the boundaries of the domain. They are identified as Korteweg-de Vries solitons with negative amplitude and subsonic velocity. The typical width and amplitude-dependent velocities are in excellent agreement with theoretical predictions based on a generalized Korteweg-de Vries equation adapted to any substrate geometry. When multiple solitons are present, they interact and form a soliton turbulencelike spectrum.
Ata-ur-Rahman,; Qamar, A.; Ali, S.; Mirza, Arshad M.
2013-04-15
We have studied the propagation of ion acoustic shock waves involving planar and non-planar geometries in an unmagnetized plasma, whose constituents are non-degenerate ultra-cold ions, relativistically degenerate electrons, and positrons. By using the reductive perturbation technique, Korteweg-deVries Burger and modified Korteweg-deVries Burger equations are derived. It is shown that only compressive shock waves can propagate in such a plasma system. The effects of geometry, the ion kinematic viscosity, and the positron concentration are examined on the ion acoustic shock potential and electric field profiles. It is found that the properties of ion acoustic shock waves in a non-planar geometry significantly differ from those in planar geometry. The present study has relevance to the dense plasmas, produced in laboratory (e.g., super-intense laser-dense matter experiments) and in dense astrophysical objects.
From Classical Nonlinear Integrable Systems to Quantum Shortcuts to Adiabaticity
NASA Astrophysics Data System (ADS)
Okuyama, Manaka; Takahashi, Kazutaka
2016-08-01
Using shortcuts to adiabaticity, we solve the time-dependent Schrödinger equation that is reduced to a classical nonlinear integrable equation. For a given time-dependent Hamiltonian, the counterdiabatic term is introduced to prevent nonadiabatic transitions. Using the fact that the equation for the dynamical invariant is equivalent to the Lax equation in nonlinear integrable systems, we obtain the counterdiabatic term exactly. The counterdiabatic term is available when the corresponding Lax pair exists and the solvable systems are classified in a unified and systematic way. Multisoliton potentials obtained from the Korteweg-de Vries equation and isotropic X Y spin chains from the Toda equations are studied in detail.
Higher-order modulation theory for resonant flow over topography
NASA Astrophysics Data System (ADS)
Daher Albalwi, M.; Marchant, T. R.; Smyth, Noel F.
2017-07-01
The flow of a fluid over isolated topography in the long wavelength, weakly nonlinear limit is considered. The upstream flow velocity is assumed to be close to a linear long wave velocity of the unforced flow so that the flow is near resonant. Higher order nonlinear, dispersive and nonlinear-dispersive terms beyond the Korteweg-de Vries approximation are included so that the flow is governed by a forced extended Korteweg-de Vries equation. Modulation theory solutions for the undular bores generated upstream and downstream of the forcing are found and used to study the influence of the higher-order terms on the resonant flow, which increases for steeper waves. These modulation theory solutions are compared with numerical solutions of the forced extended Korteweg-de Vries equation for the case of surface water waves. Good comparison is obtained between theoretical and numerical solutions, for properties such as the upstream and downstream solitary wave amplitudes and the widths of the bores.
A scattering view of the Bogoliubov-de Gennes equations
Simonucci, Stefano; Garberoglio, Giovanni; Taioli, Simone
2012-09-26
We advocate the use of the T -matrix of the pair potential to study the properties of ultracold Fermi gases in the mean-field approximation. Our approach does not require renormalization procedures even in the limit of contact interaction, and it provides a rigorous definition of the range of the potential. We also rewrite the Bogoliubov-de Gennes equation for the pairing function as a function of the T-matrix, and use it to investigate finite-range effects on the main thermodynamic observables in a gas of {sup 6}Li atoms at unitarity, calculating the pair potential with ab initio quantum chemical methods.
Structure of internal solitary waves in two-layer fluid at near-critical situation
NASA Astrophysics Data System (ADS)
Kurkina, O.; Singh, N.; Stepanyants, Y.
2015-05-01
A new model equation describing weakly nonlinear long internal waves at the interface between two thin layers of different density is derived for the specific relationships between the densities, layer thicknesses and surface tension between the layers. The equation derived and dubbed here the Gardner-Kawahara equation represents a natural generalisation of the well-known Korteweg-de Vries (KdV) equation containing the cubic nonlinear term as well as fifth-order dispersion term. Solitary wave solutions are investigated numerically and categorised in terms of two dimensionless parameters, the wave speed and fifth-order dispersion. The equation derived may be applicable to wave description in other media.
1986-04-08
equation may be thought of as the singular integral form of the Korteweg - deVries equation ut + 2uu +u =0. (4) A preprint is almost ready on this work...singular integral equation since the well known Benjamin-Ono equation ( solved by us in 1983): ut + 2uu + Hu 0. (3) It should be noted that the Benjamin-Ono...Secondly, we have recently solved an n-dimensional generalization of the sine-Gordon equation which had been studied earlier and derived by a group of
Discrete reductive perturbation technique
Levi, Decio; Petrera, Matteo
2006-04-15
We expand a partial difference equation (P{delta}E) on multiple lattices and obtain the P{delta}E which governs its far field behavior. The perturbative-reductive approach is here performed on well-known nonlinear P{delta}Es, both integrable and nonintegrable. We study the cases of the lattice modified Korteweg-de Vries (mKdV) equation, the Hietarinta equation, the lattice Volterra-Kac-Van Moerbeke equation and a nonintegrable lattice KdV equation. Such reductions allow us to obtain many new P{delta}Es of the nonlinear Schroedinger type.
NASA Astrophysics Data System (ADS)
Ayari, Mohamed Ali
Le present travail est une contribution a l'etude des equations aux derivees partielles a valeurs de Grassmann (superequations) qui constituent une extension des equations usuelles en ce sens qu'elles contiennent en plus des variables dependantes et independantes anticommutantes (fermioniques). Nous nous sommes d'abord poses la question de savoir comment generaliser la notion de groupe de Lie de symetrie de telles superequations. En effet, de telles equations etant souvent supersymetriques, elles englobent une symetrie plus large que les equations usuelles. Ensuite, nous avons utilise la methode de reduction par symetrie pour donner des solutions explicites. Nous nous sommes attardes plus specifiquement sur la superequation de KdV (N = 2) qui apparai t comme la version supersymetrique des equations de KdV et KdV modifiee. Calculer la superalgebre de Lie de symetrie d'un supersysteme est une tache tres lourde qui depend de l'ordre et du nombre de variables dependantes et independantes. Afin de surmonter ce probleme algorithmique, nous avons realise une premiere version d'un code appele GLie permettant non seulement le calcul des superequations determinantes des systemes d'equations differentielles a valeurs de Grassmann mais aussi des equations aux derivees partielles usuelles. Ce programme ecrit dans le langage Maple permet d'economiser le temps et d'echapper aux inevitables erreurs de calculs.
Propagation of ultrashort polarized light pulses in a nonlinear medium
Maimistov, A.I.
1995-03-01
Propagation of ultrashort optical pulses in a medium with degenerate resonance levels with respect to the angular momentum projections is considered. Under the assumption that the Rabi frequency is much smaller than the transition frequency and without using the slowly varying envelope approximation, a new nonlinear equation is obtained for describing this pulse dynamics. In the particular case when the pulse polarization is not changed, this is the modified Korteweg-de Vries equation. In the approximation of slowly varying envelopes, the reduced wave equation transforms into the vector nonlinear Schroedinger equation. 13 refs.
NASA Astrophysics Data System (ADS)
Haider, Md. Masum
2016-12-01
An attempt has been taken to find a general equation for degenerate pressure of Chandrasekhar and constants, by using which one can study nonrelativistic as well as ultra-relativistic cases instead of two different equations and constants. Using the general equation, ion-acoustic solitary and shock waves have been studied and compared, numerically and graphically, the two cases in same situation of electron-positron-ion plasmas. Korteweg-de Vries (KdV) and KdV-Barger equations have been derived as well as their solution to study the soliton and shock profiles, respectively.
2006-09-30
αηηx + βη = 0 (1) where co = gh , α = 3co / 2h and . The KdV equation has the generalized Fourier solution (for periodic and/or quasi... numerical integration of the partial differential equations of surface water waves is the long-term goal of this work. The approach is a...applications of the method. APPROACH We first consider the shallow water equation known as the Korteweg-deVries ( KdV ) equation ): ηt + coηx
The Magnetic Effects of Shallow Water Internal Solitons
1986-03-01
The zeroth order equation for the horizontal structure function is the Korteweg - deVries equation Whitham. 1974 i,• c Oa- 0,7 - -Y a o ) (4 The...the Maxwell equations of electromagnetism. The results indicate that the spectral levels are fairly high in the ocean’s inte- "rior, but boundary...to unit). perturbation methods e.g., Benny. 1966: Whitham. 1974 may be used 2. to get relative]% simple equations for i? and o- The lowest order
A non-standard Lax formulation of the Harry Dym hierarchy and its supersymmetric extension
NASA Astrophysics Data System (ADS)
Tian, Kai; Popowicz, Ziemowit; Liu, Q. P.
2012-03-01
For the Harry Dym hierarchy, a non-standard Lax formulation is deduced from that of the Korteweg-de Vries (KdV) equation through a reciprocal transformation. By supersymmetrizing this Lax operator, a new N = 2 supersymmetric extension of the Harry Dym hierarchy is constructed, and is further shown to be linked to one of the N = 2 supersymmetric KdV equations through the superconformal transformation. The bosonic limit of this new N = 2 supersymmetric Harry Dym equation is related to a coupled system of KdV-MKdV equations.
Long-Term VRI Photometry of ρ Cassiopeiae
NASA Astrophysics Data System (ADS)
Percy, John R.; Kolin, David L.; Henry, Gregory W.
2000-03-01
We report over 5700 days (15 years) of VRI photometry of the yellow hypergiant variable star ρ Cassiopeiae. The V-I color curve is generally in phase with the V light curve on timescales of a few hundred days, but there is a 4000 day variation in V-I which is absent from the light curve. The approximate ratio of Δ(V-I)/ΔV is 0.46. The most conspicuous period in the light curve, in the autocorrelation diagram, and in the power spectrum is about 820 days. Less significant periods of 380, 510, and 645 days also appear in the power spectrum, and there are many subcycles in the light curve with lengths of 200-500 days. Since the most recent comprehensive analysis of the light curve of ρ Cas found a dominant period of 300 days, we conclude that the behavior of the star is quite variable with time.
NASA Astrophysics Data System (ADS)
Tamilselvan, K.; Kanna, T.; Khare, Avinash
2017-10-01
We systematically construct a distinct class of complex potentials including parity-time (PT ) symmetric potentials for the stationary Schrödinger equation by using the soliton and periodic solutions of the four integrable real nonlinear evolution equations (NLEEs), namely the sine-Gordon (sG) equation, the modified Korteweg–de Vries (mKdV) equation, combined mKdV–sG equation and the Gardner equation. These potentials comprise of kink, breather, bion, elliptic bion, periodic and soliton potentials which are explicitly constructed from the various respective solutions of the NLEEs. We demonstrate the relevance between the identified complex potentials and the potential of the graphene model from an application point of view.
Automatic Processing of Digital Ionograms and Full Wave Solutions for the Profile Inversion Problem.
1981-11-01
Korteweg - deVries Equation ," J. Math. Phys., 18, 2445 (1977). Kay, I., "The Inverse Scattering Problem," Report No. EM-74 of the Institute of Mathematical...3.2 Comparison of the IWKB Method with the Full-Wave Method for Profiles for Which the Full-Wave Equation can be Solved for Exactly 45 3.2.1 General...Section 2 describes the automatic scaling of Digisonde ionograms, and Section 3 investigates the possibility of solving the Schroedinger wave equation for
Dust acoustic waves in an inhomogeneous plasma having dust size distribution
NASA Astrophysics Data System (ADS)
Banerjee, Gadadhar; Maitra, Sarit
2017-07-01
Propagations of nonlinear dust acoustic solitary waves in an inhomogeneous unmagnetized dusty plasma having power law dust distribution are investigated. Using a reductive perturbation technique, a variable coefficient deformed Korteweg-deVries (VCdKdV) equation is derived from the basic set of hydrodynamic equations. The generalized expansion method is employed to obtain a solitary wave solution for the VCdKdV equation. The different propagation characteristics of the solitary waves are studied in the presence of both plasma inhomogeneity and dust distribution.
Relative Energy for the Korteweg Theory and Related Hamiltonian Flows in Gas Dynamics
NASA Astrophysics Data System (ADS)
Giesselmann, Jan; Lattanzio, Corrado; Tzavaras, Athanasios E.
2017-03-01
We consider a Euler system with dynamics generated by a potential energy functional. We propose a form for the relative energy that exploits the variational structure and we derive a relative energy identity. When applied to specific energies, this yields relative energy identities for the Euler-Korteweg, the Euler-Poisson, the Quantum Hydrodynamics system, and low order approximations of the Euler-Korteweg system. For the Euler-Korteweg system we prove a stability theorem between a weak and a strong solution and an associated weak-strong uniqueness theorem. In the second part we focus on the Navier-Stokes-Korteweg system (NSK) with non-monotone pressure laws, and prove stability for the NSK system via a modified relative energy approach. We prove the continuous dependence of solutions on initial data and the convergence of solutions of a low order model to solutions of the NSK system. The last two results provide physically meaningful examples of how higher order regularization terms enable the use of the relative energy framework for models with energies which are not poly- or quasi-convex, compensated by higher-order gradients.
Data-driven discovery of partial differential equations
Rudy, Samuel H.; Brunton, Steven L.; Proctor, Joshua L.; Kutz, J. Nathan
2017-01-01
We propose a sparse regression method capable of discovering the governing partial differential equation(s) of a given system by time series measurements in the spatial domain. The regression framework relies on sparsity-promoting techniques to select the nonlinear and partial derivative terms of the governing equations that most accurately represent the data, bypassing a combinatorially large search through all possible candidate models. The method balances model complexity and regression accuracy by selecting a parsimonious model via Pareto analysis. Time series measurements can be made in an Eulerian framework, where the sensors are fixed spatially, or in a Lagrangian framework, where the sensors move with the dynamics. The method is computationally efficient, robust, and demonstrated to work on a variety of canonical problems spanning a number of scientific domains including Navier-Stokes, the quantum harmonic oscillator, and the diffusion equation. Moreover, the method is capable of disambiguating between potentially nonunique dynamical terms by using multiple time series taken with different initial data. Thus, for a traveling wave, the method can distinguish between a linear wave equation and the Korteweg–de Vries equation, for instance. The method provides a promising new technique for discovering governing equations and physical laws in parameterized spatiotemporal systems, where first-principles derivations are intractable. PMID:28508044
Viscosity-dependent inertial spectra of the Burgers and Korteweg–deVries–Burgers equations
Chorin, Alexandre J.; Hald, Ole H.
2005-01-01
We show that the inertial range spectrum of the Burgers equation has a viscosity-dependent correction at any wave number when the viscosity is small but not zero. We also calculate the spectrum of the Korteweg–deVries–Burgers equation and show that it can be partially mapped onto the inertial spectrum of a Burgers equation with a suitable effective diffusion coefficient. These results are significant for the understanding of turbulence. PMID:15753299
Nonlinear dust-acoustic solitary waves and shocks in dusty plasmas with a pair of trapped ions
NASA Astrophysics Data System (ADS)
Adhikary, Nirab C.; Misra, Amar P.; Deka, Manoj K.; Dev, Apul N.
2017-07-01
The propagation characteristics of small-amplitude dust-acoustic (DA) solitary waves (SWs) and shocks are studied in an unmagnetized dusty plasma with a pair of trapped positive and negative ions. Using the standard reductive perturbation technique with two different scalings of stretched coordinates, the evolution equations for DA SWs and shocks are derived in the form of complex Korteweg-de Vries and Burgers' equations. The effects of the dust charge variation, the dust thermal pressure, and the ratios of the positive to negative ion number densities as well as the free to trapped ion temperatures on the profiles of SWs and shocks are analysed and discussed.
Propagation of dust acoustic solitary waves in inhomogeneous plasma with dust charge fluctuations
NASA Astrophysics Data System (ADS)
Gogoi, L. B.; Deka, P. N.
2017-03-01
Propagations of dust acoustic solitary waves are theoretically investigated in a collisionless, unmagnetized weakly inhomogeneous plasma. The plasma that is considered here consists of negatively charged dust grains and Boltzmann distributed electrons and ions in the presence of dust charge fluctuations. The fluid equations that we use for description of such plasmas are reduced to a modified Korteweg-de-Vries equation by employing a reductive perturbation method. In this investigation, we have used space-time stretched coordinates appropriate for the inhomogeneous plasmas. From the numerical results, we have observed a significant influence of inhomogeneity parameters on the propagation of dust acoustic solitary waves.
Wave Kinematics and Sediment Suspension at Wave Breaking Point.
1982-06-01
is the fall velocity in oscillatory flow and W is the amplitude I of vertical velocity component. Equating Eqs. (3-30) I and (3-31) and solving the w...of the waves. The I cnoidal wave model was developed by Korteweg and DeVries (1895).. At the limits, the cnoidal wave approaches the I I I I I 119I I...experimental data. At present, sediment suspension in a fluid media Lis treated as a diffusion-dispersion process, and the [governing equation takes the
Nonlinear waves in dense dusty plasmas with high fugacity
NASA Astrophysics Data System (ADS)
Rao, N. N.; Shukla, P. K.
2001-01-01
Nonlinear propagation of small, but finite, amplitude electrostatic dust waves has been investigated in the low as well as high fugacity regimes by deriving the corresponding Boussinesq equation which, for unidirectional propagation, reduces to the Korteweg-de Vries equation. The dust-acoustic wave (DAW) solitons are shown to correspond to the tenuous (low fugacity) dusty plasmas, while in the dense (high fugacity) regime the solitons are associated with the dust-Coulomb waves (DCWs). Unlike the DAW solitons which are (dust) density compressional and supersonic, the DCW solitons are (dust) density rarefactive and propagate with super-Coulombic speeds.
Soliton splitting in quenched classical integrable systems
NASA Astrophysics Data System (ADS)
Gamayun, O.; Semenyakin, M.
2016-08-01
We take a soliton solution of a classical non-linear integrable equation and quench (suddenly change) its non-linearity parameter. For that we multiply the amplitude or the width of a soliton by a numerical factor η and take the obtained profile as a new initial condition. We find the values of η for which the post-quench solution consists of only a finite number of solitons. The parameters of these solitons are found explicitly. Our approach is based on solving the direct scattering problem analytically. We demonstrate how it works for Korteweg-de Vries, sine-Gordon and non-linear Schrödinger integrable equations.
Effect of trapped electrons on soliton propagation in a plasma having a density gradient
Aziz, Farah; Stroth, Ulrich
2009-03-15
A Korteweg-deVries equation with an additional term due to the density gradient is obtained using reductive perturbation technique in an unmagnetized plasma having a density gradient, finite temperature ions, and two-temperature nonisothermal (trapped) electrons. This equation is solved to get the solitary wave solution using sine-cosine method. The phase velocity, soliton amplitude, and width are examined under the effect of electron and ion temperatures and their concentrations. The effect of ion (electron) temperature is found to be more significant in the presence of larger (smaller) number of trapped electrons in the plasma.
Michev, Iordan P.
2006-09-15
In the first part of this paper we consider the transformation of the cubic identities for general Korteweg-de Vries (KdV) tau functions from [Mishev, J. Math. Phys. 40, 2419-2428 (1999)] to the specific identities for trigonometric KdV tau functions. Afterwards, we consider the Fay identity as a functional equation and provide a wide set of solutions of this equation. The main result of this paper is Theorem 3.4, where we generalize the identities from Mishev. An open problem is the transformation of the cubic identities from Mishev to the specific identities for elliptic KdV tau functions.
Higher order solutions to ion-acoustic solitons in a weakly relativistic two-fluid plasma
Gill, Tarsem Singh; Bala, Parveen; Kaur, Harvinder
2008-12-15
The nonlinear wave structure of small amplitude ion-acoustic solitary waves (IASs) is investigated in a two-fluid plasma consisting of weakly relativistic streaming ions and electrons. Using the reductive perturbation theory, the basic set of governing equations is reduced to the Korteweg-de Vries (KdV) equation for the lowest order perturbation. This analysis is further extended using the renormalization technique for the inclusion of higher order nonlinear and dispersive effects for better accuracy. The effect of higher order correction and various parameters on the soliton characteristics is investigated and also discussed.
Nonlinear wave propagation in a strongly coupled collisional dusty plasma.
Ghosh, Samiran; Gupta, Mithil Ranjan; Chakrabarti, Nikhil; Chaudhuri, Manis
2011-06-01
The propagation of a nonlinear low-frequency mode in a strongly coupled dusty plasma is investigated using a generalized hydrodynamical model. For the well-known longitudinal dust acoustic mode a standard perturbative approach leads to a Korteweg-de Vries (KdV) soliton. The strong viscoelastic effect, however, introduced a nonlinear forcing and a linear damping in the KdV equation. This novel equation is solved analytically to show a competition between nonlinear forcing and dissipative damping. The physical consequence of such a solution is also sketched.
Sah, O.P.; Goswami, K.S. )
1994-10-01
Considering an unmagnetized plasma consisting of relativistic drifting electrons and nondrifting thermal ions and by using reductive perturbation method, a usual Korteweg--de Vries (KdV) equation and a generalized form of KdV equation are derived. It is found that while the former governs the dynamics of a small-amplitude rarefactive modified electron acoustic (MEA) soliton, the latter governs the dynamics of a weak compressive modified electron acoustic double layer. The influences of relativistic effect on the propagation of such a soliton and double layer are examined. The relevance of this investigation to space plasma is pointed out.
Nonlinear wave propagation in a strongly coupled collisional dusty plasma
Ghosh, Samiran; Gupta, Mithil Ranjan; Chakrabarti, Nikhil; Chaudhuri, Manis
2011-06-15
The propagation of a nonlinear low-frequency mode in a strongly coupled dusty plasma is investigated using a generalized hydrodynamical model. For the well-known longitudinal dust acoustic mode a standard perturbative approach leads to a Korteweg-de Vries (KdV) soliton. The strong viscoelastic effect, however, introduced a nonlinear forcing and a linear damping in the KdV equation. This novel equation is solved analytically to show a competition between nonlinear forcing and dissipative damping. The physical consequence of such a solution is also sketched.
Wakes and precursor soliton excitations by a moving charged object in a plasma
Kumar Tiwari, Sanat; Sen, Abhijit
2016-02-15
We study the evolution of nonlinear ion acoustic wave excitations due to a moving charged source in a plasma. Our numerical investigations of the full set of cold fluid equations go beyond the usual weak nonlinearity approximation and show the existence of a rich variety of solutions including wakes, precursor solitons, and “pinned” solitons that travel with the source velocity. These solutions represent a large amplitude generalization of solutions obtained in the past for the forced Korteweg deVries equation and can find useful applications in a variety of situations in the laboratory and in space, wherever there is a large relative velocity between the plasma and a charged object.
Maitra, Sarit; Banerjee, Gadadhar
2014-11-15
The influence of dust size distribution on the dust ion acoustic solitary waves in a collisional dusty plasma is investigated. It is found that dust size distribution changes the amplitude and width of a solitary wave. A critical wave number is derived for the existence of purely damping mode. A deformed Korteweg-de Vries (dKdV) equation is obtained for the propagation of weakly nonlinear dust ion acoustic solitary waves and the effect of different plasma parameters on the solution of this equation is also presented.
Shock wave in magnetized dusty plasmas with dust charging and nonthermal ion effects
Zhang Liping; Xue Jukui
2005-04-15
The effects of the external magnetized field, nonadiabatic dust charge fluctuation, and nonthermally distributed ions on three-dimensional dust acoustic shock wave in dusty plasmas have been investigated. By using the reductive perturbation method, a Korteweg-de Vries (KdV) Burger equation governing the dust acoustic shock wave is derived. The results of numerical integrations of KdV Burger equation show that the external magnetized field, nonthermally distributed ions, and nonadiabatic dust charge fluctuation have strong influence on the shock structures.
Ion acoustic shocks in magneto rotating Lorentzian plasmas
Hussain, S.; Akhtar, N.; Hasnain, H.
2014-12-15
Ion acoustic shock structures in magnetized homogeneous dissipative Lorentzian plasma under the effects of Coriolis force are investigated. The dissipation in the plasma system is introduced via dynamic viscosity of inertial ions. The electrons are following the kappa distribution function. Korteweg-de Vries Burger (KdVB) equation is derived by using reductive perturbation technique. It is shown that spectral index, magnetic field, kinematic viscosity of ions, rotational frequency, and effective frequency have significant impact on the propagation characteristic of ion acoustic shocks in such plasma system. The numerical solution of KdVB equation is also discussed and transition from oscillatory profile to monotonic shock for different plasma parameters is investigated.
NASA Astrophysics Data System (ADS)
Simonucci, S.; Strinati, G. C.
2014-02-01
We derive a nonlinear differential equation for the gap parameter of a superfluid Fermi system by performing a suitable coarse graining of the Bogoliubov-de Gennes (BdG) equations throughout the BCS-BEC crossover, with the aim of replacing the time-consuming solution of the original BdG equations by the simpler solution of this novel equation. We perform a favorable numerical test on the validity of this new equation over most of the temperature-coupling phase diagram, by an explicit comparison with the full solution of the original BdG equations for an isolated vortex. We also show that the new equation reduces both to the Ginzburg-Landau equation for Cooper pairs in weak coupling close to the critical temperature and to the Gross-Pitaevskii equation for composite bosons in strong coupling at low temperature.
Phase-Field and Korteweg-Type Models for the Time-Dependent Flow of Compressible Two-Phase Fluids
NASA Astrophysics Data System (ADS)
Freistühler, Heinrich; Kotschote, Matthias
2017-04-01
Various versions of the Navier-Stokes-Allen-Cahn (NSAC), the Navier-Stokes-Cahn-Hilliard (NSCH), and the Navier-Stokes-Korteweg (NSK) equations have been used in the literature to model the dynamics of two-phase fluids. One main purpose of this paper consists in (re-)deriving NSAC, NSCH and NSK from first principles, in the spirit of rational mechanics, for fluids of very general constitutive laws. For NSAC, this deduction confirms and extends a proposal of Blesgen. Regarding NSCH, it continues work of Lowengrub and Truskinovsky and provides the apparently first justified formulation in the non-isothermal case. For NSK, it yields a most natural correction to the formulation by Dunn and Serrin. The paper uniformly recovers as examples various classes of fluids, distinguished according to whether none, one, or both of the phases are compressible, and according to the nature of their co-existence. The latter is captured not only by the mixing energy, but also by a `mixing rule'—a constitutive law that characterizes the type of the mixing. A second main purpose of the paper is to communicate the apparently new observation that in the case of two immiscible incompressible phases of different temperature-independent specific volumes, NSAC reduces literally to NSK. This finding may be considered as an independent justification of NSK. An analogous fact is shown for NSCH, which under the same assumption reduces to a new non-local version of NSK.
NASA Astrophysics Data System (ADS)
Masood, W.; Hamid, Naira; Ilyas, Iffat; Siddiq, M.
2017-06-01
In this paper, we have investigated electrostatic solitary and shock waves in an unmagnetized relativistic electron-ion (ei) plasma in the presence of warm ions and trapped electrons. In this regard, we have derived the trapped Korteweg-de Vries Burgers (TKdVB) equation using the small amplitude approximation method, which to the best of our knowledge has not been investigated in plasmas. Since the TKdVB equation involves fractional nonlinearity on account of trapped electrons, we have employed a smartly crafted extension of the tangent hyperbolic method and presented the solution of the TKdVB equation in this paper. The limiting cases of the TKdVB equation yield trapped Burgers (TB) and trapped Korteweg-de Vries (TKdV) equations. We have also presented the solutions of TB and TKdV equations. We have also explored how the plasma parameters affect the propagation characteristics of the nonlinear structures obtained for these modified nonlinear partial differential equations. We hope that the present work will open new vistas of research in the nonlinear plasma theory both in classical and quantum plasmas.
One particle quantum equation in a de Sitter spacetime
NASA Astrophysics Data System (ADS)
Frick, R. A.
2014-08-01
We consider a free particle in a de Sitter spacetime. We use a picture in which the analogs of the Schr\\"odinger operators of the particle are independent of both the time and the space coordinates. These operators induce operators which are related to Killing vectors of the de Sitter spacetime.
Wheeler-DeWitt equation and Feynman diagrams
NASA Astrophysics Data System (ADS)
Barvinsky, Andrei O.; Kiefer, Claus
1998-08-01
We present a systematic expansion of all constraint equations in canonical quantum gravity up to the order of the inverse Planck mass squared. It is demonstrated that this method generates the Feynman diagrammatic technique involving graviton loops and vertices. It also reveals explicitly the back-reaction effects of quantized matter and graviton vacuum polarization. This provides an explicit correspondence between the frameworks of canonical and covariant quantum gravity in the semiclassical limit.
NASA Astrophysics Data System (ADS)
Demiray, Hilmi; Bayındır, Cihan
2015-09-01
In the present work, we consider the propagation of nonlinear electron-acoustic non-planar waves in a plasma composed of a cold electron fluid, hot electrons obeying a trapped/vortex-like distribution, and stationary ions. The basic nonlinear equations of the above described plasma are re-examined in the cylindrical coordinates through the use reductive perturbation method in the long-wave approximation. The modified cylindrical Korteweg-de Vries equation with fractional power nonlinearity is obtained as the evolution equation. Due to the nature of nonlinearity, which is fractional, this evolution equation cannot be reduced to the conventional Korteweg-de Vries equation. An analytical solution to the evolution equation, by use of the method developed by Demiray [Appl. Math. Comput. 132, 643 (2002); Comput. Math. Appl. 60, 1747 (2010)] and a numerical solution by employing a spectral scheme are presented and the results are depicted in a figure. The numerical results reveal that both solutions are in good agreement.
Dispersive shock waves in nematic liquid crystals
NASA Astrophysics Data System (ADS)
Smyth, Noel F.
2016-10-01
The propagation of coherent light with an initial step intensity profile in a nematic liquid crystal is studied using modulation theory. The propagation of light in a nematic liquid crystal is governed by a coupled system consisting of a nonlinear Schrödinger equation for the light beam and an elliptic equation for the medium response. In general, the intensity step breaks up into a dispersive shock wave, or undular bore, and an expansion fan. In the experimental parameter regime for which the nematic response is highly nonlocal, this nematic bore is found to differ substantially from the standard defocusing nonlinear Schrödinger equation structure due to the effect of the nonlocality of the nematic medium. It is found that the undular bore is of Korteweg-de Vries equation-type, consisting of bright waves, rather than of nonlinear Schrödinger equation-type, consisting of dark waves. In addition, ahead of this Korteweg-de Vries bore there can be a uniform wavetrain with a short front which brings the solution down to the initial level ahead. It is found that this uniform wavetrain does not exist if the initial jump is below a critical value. Analytical solutions for the various parts of the nematic bore are found, with emphasis on the role of the nonlocality of the nematic medium in shaping this structure. Excellent agreement between full numerical solutions of the governing nematicon equations and these analytical solutions is found.
The Wheeler-DeWitt Equation in Filćhenkov Model: The Lie Algebraic Approach
NASA Astrophysics Data System (ADS)
Panahi, H.; Zarrinkamar, S.; Baradaran, M.
2016-11-01
The Wheeler-DeWitt equation in Filćhenkov model with terms related to strings, dust, relativistic matter, bosons and fermions, and ultra stiff matter is solved in a quasi-exact analytical manner via the Lie algebraic approach. In the calculations, using the representation theory of sl(2), the general (N+1)-dimensional matrix equation is constructed whose determinant yields the solutions of the problem.
Axial and polar gravitational wave equations in a de Sitter expanding universe by Laplace transform
NASA Astrophysics Data System (ADS)
Viaggiu, Stefano
2017-02-01
In this paper we study the propagation in a de Sitter universe of gravitational waves generated by perturbating some unspecified spherical astrophysical object in the frequencies domain. We obtain the axial and polar perturbation equations in a cosmological de Sitter universe in the usual comoving coordinates, the coordinates we occupy in our galaxy. We write down the relevant equations in terms of Laplace transform with respect to the comoving time t instead of the usual Fourier one that is no longer available in a cosmological context. Both axial and polar perturbation equations are expressed in terms of a non trivial mixture of retarded-advanced metric coefficients with respect to the Laplace parameter s (complex translation). The axial case is studied in more detail. In particular, the axial perturbations can be reduced to a master linear second-order differential equation in terms of the Regge-Wheeler function Z where a coupling with a retarded Z with respect to the cosmological time t is present. It is shown that a de Sitter expanding universe can change the frequency ω of a gravitational wave as perceived by a comoving observer. The polar equations are much more involved. Nevertheless, we show that the polar perturbations can also be expressed in terms of four independent integrable differential equations.
Theoretical analysis of the density wave in a new continuum model and numerical simulation
NASA Astrophysics Data System (ADS)
Lai, Ling-Ling; Cheng, Rong-Jun; Li, Zhi-Peng; Ge, Hong-Xia
2014-05-01
Considered the effect of traffic anticipation in the real world, a new anticipation driving car following model (AD-CF) was proposed by Zheng et al. Based on AD-CF model, adopted an asymptotic approximation between the headway and density, a new continuum model is presented in this paper. The neutral stability condition is obtained by applying the linear stability theory. Additionally, the Korteweg-de Vries (KdV) equation is derived via nonlinear analysis to describe the propagating behavior of traffic density wave near the neutral stability line. The numerical simulation and the analytical results show that the new continuum model is capable of explaining some particular traffic phenomena.
A new car-following model with two delays
NASA Astrophysics Data System (ADS)
Yu, Lei; Shi, Zhong-ke; Li, Tong
2014-01-01
A new car-following model is proposed by taking into account two different time delays in sensing headway and velocity. The effect of time delays on the stability analysis is studied. The theoretical and numerical results show that traffic jams are suppressed efficiently when the difference between two time delays decreases and those can be described by the solution of the modified Korteweg-de Vries (mKdV) equation. Traffic flow is more stable with two delays in headway and velocity than in the case with only one delay in headway. The impact of local small disturbance to the system is also studied.
A novel car following model considering average speed of preceding vehicles group
NASA Astrophysics Data System (ADS)
Sun, Dihua; Kang, Yirong; Yang, Shuhong
2015-10-01
In this paper, a new car following model is presented by considering the average speed effect of preceding vehicles group in cyber-physical systems (CPS) environment. The effect of this new consideration upon the stability of traffic flow is examined through linear stability analysis. A modified Korteweg-de Vries (mKdV) equation was derived via nonlinear analysis to describe the propagating behavior of traffic density wave near the critical point. Good agreement between the simulation and the analytical results shows that average speed of preceding vehicles group leads to the stabilization of traffic systems, and thus can efficiently suppress the emergence of traffic jamming.
Head-on collision of dust-ion-acoustic soliton in quantum pair-ion plasma
Chatterjee, Prasanta; Ghorui, Malay kr.; Wong, C. S.
2011-10-15
In this paper, we study the head-on collision between two dust ion acoustic solitons in quantum pair-ion plasma. Using the extended Poincare-Lighthill-Kuo method, we obtain the Korteweg-de Vries equation, the phase shifts, and the trajectories after the head-on collision of the two dust ion acoustic solitons. It is observed that the phase shifts are significantly affected by the values of the quantum parameter H, the ratio of the multiples of the charge state and density of positive ions to that of the negative ions {beta} and the concentration of the negatively charged dust particles {delta}.
NASA Astrophysics Data System (ADS)
Choudhury, Sourav; Das, Tushar Kanti; Ghorui, Malay Kr.; Chatterjee, Prasanta
2017-06-01
Collisions of solitary pulses in a four species quantum semiconductor plasma consisting of degenerate electrons, degenerate holes, and non-degenerate ions are investigated. The electron and hole exchange-correlation forces between the identical particles when their wave functions overlap due to the high number densities are considered. Using the extended Poincarê-Lighthill-Kue method in opposite directions, two Korteweg-de Vries equations are derived. Hirota's method is used to derive the analytical phase shifts after the collision of one soliton and two soliton. Typical values for GaAs, GaSb, GaN, and InP semiconductors are considered to analyze the effects after collisions.
Interaction of fast magnetoacoustic solitons in dense plasmas
Jahangir, R.; Saleem, Khalid; Masood, W.; Siddiq, M.; Batool, Nazia
2015-09-15
One dimensional propagation of fast magnetoacoustic solitary waves in dense plasmas with degenerate electrons is investigated in this paper in the small amplitude limit. In this regard, Korteweg deVries equation is derived and discussed using the plasma parameters that are typically found in white dwarf stars. The interaction of fast magnetoacoustic solitons is explored by using the Hirota bilinear formalism, which admits multi soliton solutions. It is observed that the values of the propagation vectors determine the interaction of solitary waves. It is further noted that the amplitude of the respective solitary waves remain unchanged after the interaction; however, they do experience a phase shift.
Weakly relativistic solitons in a cold plasma with electron inertia
Kalita, B.C.; Barman, S.N.; Goswami, G.
1996-01-01
Ion-acoustic solitons have been investigated in a cold plasma in the presence of electron inertia through the derivation of the Korteweg{endash}de Vries (KdV) equation taking into account of weakly relativistic effects. Interestingly, relativistic solitons of both compressive and rarefactive characters are found to exist at the negligible difference of {ital u}{sub 0}/{ital c} and {ital v}{sub 0}/{ital c} ({ital u}{sub 0}, {ital v}{sub 0} being the initial speeds of streaming electrons and ions respectively, and {ital c}, the velocity of light) of the order 1{times}10{sup -7}. {copyright} {ital 1996 American Institute of Physics.}
Plasma shock waves excited by THz radiation
NASA Astrophysics Data System (ADS)
Rudin, S.; Rupper, G.; Shur, M.
2016-10-01
The shock plasma waves in Si MOS, InGaAs and GaN HEMTs are launched at a relatively small THz power that is nearly independent of the THz input frequency for short channel (22 nm) devices and increases with frequency for longer (100 nm to 1 mm devices). Increasing the gate-to-channel separation leads to a gradual transition of the nonlinear waves from the shock waves to solitons. The mathematics of this transition is described by the Korteweg-de Vries equation that has the single propagating soliton solution.
Fedila, D. Ali; Djebli, M.
2010-10-15
The effect of collision on small amplitude dust-acoustic waves is investigated for a plasma with positively charged dust grains. Taking into account the presence of different electron populations in thermal equilibrium, a modified Korteweg-de Vries equation is established. The existence conditions and nature of the waves, i.e., rarefactive or compressive, are found to be mainly dependent on the temperature and the density of the cold electrons. The present model is used to understand the salient features of the fully nonlinear dust-acoustic waves in the lower region of the Earth's ionosphere, at an altitude of {approx}85 km with the presence of an external heating source.
Nonplanar waves with electronegative dusty plasma
Zobaer, M. S.; Mukta, K. N.; Nahar, L.; Mamun, A. A.; Roy, N.
2013-04-15
A rigorous theoretical investigation has been made of basic characteristics of the nonplanar dust-ion-acoustic shock and solitary waves in electronegative dusty plasma containing Boltzmann electrons, Boltzmann negative ions, inertial positive ions, and charge fluctuating (negatively charged) stationary dust. The Burgers' and Korteweg-de Vries (K-dV) equations, which is derived by reductive perturbation technique, is numerically solved to examine the effects of nonplanar geometry on the basic features of the DIA shock and solitary waves formed in the electronegative dusty plasma. The implications of the results (obtained from this investigation) in space and laboratory experiments are briefly discussed.
Magnetosonic wave in pair-ion electron collisional plasmas
NASA Astrophysics Data System (ADS)
Hussain, S.; Hasnain, H.
2017-03-01
Low frequency magnetosonic waves in positive and negative ions of equal mass and opposite charges in the presence of electrons in collisional plasmas are studied. The collisions of ions and electrons with neutrals are taken into account. The nonlinearities in the plasma system arise due to ion and electrons flux, Lorentz forces, and plasma current densities. The reductive perturbation method is applied to derive the Damped Korteweg de Vries (DKdV) equation. The time dependent solution of DKdV is presented. The effects of variations of different plasma parameters on propagation characteristics of magnetosonic waves in pair-ion electron plasma in the context of laboratory plasmas are discussed.
Rarefaction solitons initiated by sheath instability
Levko, Dmitry
2015-09-15
The instability of the cathode sheath initiated by the cold energetic electron beam is studied by the one-dimensional fluid model. Numerical simulations show the generation of travelling rarefaction solitons at the cathode. It is obtained that the parameters of these solitons strongly depend on the parameters of electron beam. The “stretched” variables are derived using the small-amplitude analysis. These variables are used in order to obtain the Korteweg-de Vries equation describing the propagation of the rarefaction solitons through the plasma with cold energetic electron beam.
Observation of axisymmetric solitary waves on the surface of a ferrofluid.
Bourdin, E; Bacri, J-C; Falcon, E
2010-03-05
We report the first observation of axisymmetric solitary waves on the surface of a cylindrical magnetic fluid layer surrounding a current-carrying metallic tube. According to the ratio between the magnetic and capillary forces, both elevation and depression solitary waves are observed with profiles in good agreement with theoretical predictions based on the magnetic analogue of the Korteweg-de Vries equation. We also report the first measurements of the velocity and the dispersion relation of axisymmetric linear waves propagating on the cylindrical ferrofluid layer that are found in good agreement with theoretical predictions.
Do the freak waves exist in soliton gas?
NASA Astrophysics Data System (ADS)
Shurgalina, Ekaterina; Pelinovsky, Efim
2016-04-01
The possibility of short-lived anomalous large waves (rogue waves) in soliton gas in the frameworks of integrable models like the Korteweg - de Vries - type equations is studied. It is shown that the dynamics of heteropolar soliton gas differs sufficiently from the dynamics of unipolar soliton fields. In particular, in the wave fields consisting of solitons with different polarities the freak wave appearance is possible. It is shown numerically in [Shurgalina and Pelinovsky, 2015]. Freak waves in the framework of the modified Korteweg-de Vries equation have been studied previously in the case of narrowband initial conditions [Grimshaw et al, 2005, 2010; Talipova, 2011]. In this case, the mechanism of freak wave generation was modulation instability of modulated quasi-sinusoidal wave packets. At the same time the modulation instability of modulated cnoidal waves was studied in the mathematical work [Driscoll & O'Neil, 1976]. Since a sequence of solitary waves can be a special case of cnoidal wave, the modulation instability can be a possible mechanism of freak wave appearance in a soliton gas. Thus, we expect that rogue wave phenomenon in soliton gas appears in nonlinear integrable models admitting an existence of modulation instability of periodic waves (like cnoidal waves). References: 1. Shurgalina E.G., Pelinovsky E.N. Dynamics of irregular wave ensembles in the coastal zone, Nizhny Novgorod State Technical University n.a. R.E. Alekseev. - Nizhny Novgorod, 2015, 179 pp. 2. Grimshaw R., Pelinovsky E., Talipova T., Sergeeva A. Rogue internal waves in the ocean: long wave model. European Physical Journal Special Topics, 2010, 185, 195 - 208. 3. Grimshaw R., Pelinovsky E., Talipova T., Ruderman M. Erdelyi R. Short-lived large-amplitude pulses in the nonlinear long-wave model described by the modified Korteweg-de Vries equation. Studied Applied Mathematics, 2005, 114 (2), 189. 4. Talipova T.G. Mechanisms of internal freak waves, Fundamental and Applied Hydrophysics
NASA Astrophysics Data System (ADS)
Javidan, Kurosh; Pakzad, Hamid Reza
2012-02-01
Propagation of cylindrical and spherical electron-acoustic solitary waves in unmagnetized plasmas consisting of cold electron fluid, hot electrons obeying a superthermal distribution and stationary ions are investigated. The standard reductive perturbation method is employed to derive the cylindrical/spherical Korteweg-de-Vries equation which governs the dynamics of electron-acoustic solitons. The effects of nonplanar geometry and superthermal hot electrons on the behavior of cylindrical and spherical electron acoustic soliton and its structure are also studied using numerical simulations.
Charging-delay induced dust acoustic collisionless shock wave: Roles of negative ions
Ghosh, Samiran; Bharuthram, R.; Khan, Manoranjan; Gupta, M. R.
2006-11-15
The effects of charging-delay and negative ions on nonlinear dust acoustic waves are investigated. It has been found that the charging-delay induced anomalous dissipation causes generation of dust acoustic collisionless shock waves in an electronegative dusty plasma. The small but finite amplitude wave is governed by a Korteweg-de Vries Burger equation in which the Burger term arises due to the charging-delay. Numerical investigations reveal that the charging-delay induced dissipation and shock strength decreases (increases) with the increase of negative ion concentration (temperature)
Tribeche, Mouloud; Bacha, Mustapha
2013-10-15
Weak dust-acoustic waves (DAWs) are addressed in a nonthermal charge varying electronegative magnetized dusty plasmas with application to the Halley Comet. A weakly nonlinear analysis is carried out to derive a Korteweg-de Vries-Burger equation. The positive ion nonthermality, the obliqueness, and magnitude of the magnetic field are found to modify the dispersive and dissipative properties of the DA shock structure. Our results may aid to explain and interpret the nonlinear oscillations that may occur in the Halley Comet Plasma.
Tribeche, Mouloud; Bacha, Mustapha
2012-12-15
The combined effects of an oblique magnetic field and electron suprathermality on weak dust-acoustic (DA) waves in a charge varying electronegative dusty plasmas with application to the Halley Comet are investigated. The correct suprathermal electron charging current is derived based on the orbit-motion limited approach. A weakly nonlinear analysis is carried out to derive a Korteweg-de Vries-Burger equation. The electron suprathermality, the obliqueness, and magnitude of the magnetic field are found to modify the dispersive properties of the DA shock structure. Our results may aid to explain and interpret the nonlinear oscillations that may occur in the Halley Comet plasma.
Nonlinear shear wave in a non Newtonian visco-elastic medium
Banerjee, D.; Janaki, M. S.; Chakrabarti, N.
2012-06-15
An analysis of nonlinear transverse shear wave has been carried out on non-Newtonian viscoelastic liquid using generalized hydrodynamic model. The nonlinear viscoelastic behavior is introduced through velocity shear dependence of viscosity coefficient by well known Carreau-Bird model. The dynamical feature of this shear wave leads to the celebrated Fermi-Pasta-Ulam problem. Numerical solution has been obtained which shows that initial periodic solutions reoccur after passing through several patterns of periodic waves. A possible explanation for this periodic solution is given by constructing modified Korteweg de Vries equation. This model has application from laboratory to astrophysical plasmas as well as in biological systems.
Multi-ion Double Layers in a Magnetized Plasma
NASA Astrophysics Data System (ADS)
Shahmansouri, M.; Alinejad, H.; Tribeche, M.
2015-11-01
A theoretical investigation is carried out to study the existence, formation and basic properties of ion acoustic (IA) double layers (DLs) in a magnetized bi-ion plasma consisting of warm/cold ions and Boltzmann distributed electrons. Based on the reductive perturbation technique, an extended Korteweg de-Vries (KdV) equation is derived. The propagation of two possible modes (fast and slow), and their evolution are investigated. The effects of obliqueness, magnitude of the magnetic field, ion concentration, polarity of ions, and ion temperature on the IA DL profile are analyzed, and then the ranges of parameters for which the IA DLs exist are investigated in details.
Imploding and exploding shocks in negative ion degenerate plasmas
Hussain, S.; Akhtar, N.
2011-08-15
Imploding and exploding shocks are studied in nonplanar geometries for negative ion degenerate plasma. Deformed Korteweg de Vries Burgers (DKdVB) equation is derived by using reductive perturbation method. Two level finite difference scheme is used for numerical analysis of DKdVB. It is observed that compressive and rarefactive shocks are observed depending on the value of quantum parameter. The effects of temperature, kinematic viscosity, mass ratio of negative to positive ions and quantum parameter on diverging and converging shocks are presented.
NASA Astrophysics Data System (ADS)
Chen, Zhengzheng; He, Lin; Zhao, Huijiang
2017-08-01
We are concerned with the construction of global smooth large-amplitude solutions to the Cauchy problem of the one-dimensional nonisothermal compressible fluid models of Korteweg type with density- and/or temperature-dependent viscosity, capillarity, and heat conductivity coefficients. Two types of global solvability results are obtained if the viscosity, capillarity, and heat conductivity coefficients satisfy some conditions, and the key point in our analysis is to deduce the positive lower and upper bounds on the density and the temperature.
Cosmological constant from a deformation of the Wheeler-DeWitt equation
NASA Astrophysics Data System (ADS)
Garattini, Remo; Faizal, Mir
2016-04-01
In this paper, we consider the Wheeler-DeWitt equation modified by a deformation of the second quantized canonical commutation relations. Such modified commutation relations are induced by a Generalized Uncertainty Principle. Since the Wheeler-DeWitt equation can be related to a Sturm-Liouville problem where the associated eigenvalue can be interpreted as the cosmological constant, it is possible to explicitly relate such an eigenvalue to the deformation parameter of the corresponding Wheeler-DeWitt equation. The analysis is performed in a Mini-Superspace approach where the scale factor appears as the only degree of freedom. The deformation of the Wheeler-DeWitt equation gives rise to a Cosmological Constant even in absence of matter fields. As a Cosmological Constant cannot exist in absence of the matter fields in the undeformed Mini-Superspace approach, so the existence of a non-vanishing Cosmological Constant is a direct consequence of the deformation by the Generalized Uncertainty Principle. In fact, we are able to demonstrate that a non-vanishing Cosmological Constant exists even in the deformed flat space. We also discuss the consequences of this deformation on the big bang singularity.
Nonlinear Laplace equation, de Sitter vacua, and information geometry
Loran, Farhang
2005-06-15
Three exact solutions say {phi}{sub 0} of massless scalar theories on Euclidean space, i.e. D=6 {phi}{sup 3}, D=4 {phi}{sup 4} and D=3 {phi}{sup 6} models are obtained which share similar properties. The information geometry of their moduli spaces coincide with the Euclidean AdS{sub 7}, AdS{sub 5} and AdS{sub 4} respectively on which {phi}{sub 0} can be described as a stable tachyon. In D=4 we recognize that the SU(2) instanton density is proportional to {phi}{sub 0}{sup 4}. The original action S[{phi}] written in terms of new scalars {phi}-tilde={phi}-{phi}{sub 0} is shown to be equivalent to an interacting scalar theory on D-dimensional de Sitter background.
Stability of Viscous St. Venant Roll Waves: From Onset to Infinite Froude Number Limit
NASA Astrophysics Data System (ADS)
Barker, Blake; Johnson, Mathew A.; Noble, Pascal; Rodrigues, L. Miguel; Zumbrun, Kevin
2017-02-01
We study the spectral stability of roll wave solutions of the viscous St. Venant equations modeling inclined shallow water flow, both at onset in the small Froude number or "weakly unstable" limit F→ 2^+ and for general values of the Froude number F, including the limit F→ +∞ . In the former, F→ 2^+, limit, the shallow water equations are formally approximated by a Korteweg-de Vries/Kuramoto-Sivashinsky (KdV-KS) equation that is a singular perturbation of the standard Korteweg-de Vries (KdV) equation modeling horizontal shallow water flow. Our main analytical result is to rigorously validate this formal limit, showing that stability as F→ 2^+ is equivalent to stability of the corresponding KdV-KS waves in the KdV limit. Together with recent results obtained for KdV-KS by Johnson-Noble-Rodrigues-Zumbrun and Barker, this gives not only the first rigorous verification of stability for any single viscous St. Venant roll wave, but a complete classification of stability in the weakly unstable limit. In the remainder of the paper, we investigate numerically and analytically the evolution of the stability diagram as Froude number increases to infinity. Notably, we find transition at around F=2.3 from weakly unstable to different, large- F behavior, with stability determined by simple power-law relations. The latter stability criteria are potentially useful in hydraulic engineering applications, for which typically 2.5≤ F≤ 6.0.
Equivalent Forms of Dirac Equations in Curved Space-times and Generalized de Broglie Relations
NASA Astrophysics Data System (ADS)
Arminjon, Mayeul; Reifler, Frank
2013-04-01
One may ask whether the relations between energy and frequency and between momentum and wave vector, introduced for matter waves by de Broglie, are rigorously valid in the presence of gravity. In this paper, we show this to be true for Dirac equations in a background of gravitational and electromagnetic fields. We first transform any Dirac equation into an equivalent canonical form, sometimes used in particular cases to solve Dirac equations in a curved space-time. This canonical form is needed to apply Whitham's Lagrangian method. The latter method, unlike the Wentzel-Kramers-Brillouin method, places no restriction on the magnitude of Planck's constant to obtain wave packets and furthermore preserves the symmetries of the Dirac Lagrangian. We show by using canonical Dirac fields in a curved space-time that the probability current has a Gordon decomposition into a convection current and a spin current and that the spin current vanishes in the Whitham approximation, which explains the negligible effect of spin on wave packet solutions, independent of the size of Planck's constant. We further discuss the classical-quantum correspondence in a curved space-time based on both Lagrangian and Hamiltonian formulations of the Whitham equations. We show that the generalized de Broglie relations in a curved space-time are a direct consequence of Whitham's Lagrangian method and not just a physical hypothesis as introduced by Einstein and de Broglie and by many quantum mechanics textbooks.
Solution of Dirac equation in Reissner-Nordström de Sitter space
NASA Astrophysics Data System (ADS)
Lyu, Yan; Cui, Song
2009-02-01
The radial parts of the Dirac equation between the outer black hole horizon and the cosmological horizon are solved in Reissner-Nordström de Sitter (RNdS) space numerically. An accurate approximation, the polynomial approximation, is used to approximate the modified tortoise coordinate \\hat r_* , which leads to the inverse function r = r(\\hat r_* ) and the potential V(\\hat r_* ). The potential V(\\hat r_* ) is replaced by a collection of step functions in sequence. Then the solution of the wave equation as well as the reflection and transmission coefficients is computed by a quantum mechanical method.
Thermal diffusion of Boussinesq solitons.
Arévalo, Edward; Mertens, Franz G
2007-10-01
We consider the problem of the soliton dynamics in the presence of an external noisy force for the Boussinesq type equations. A set of ordinary differential equations (ODEs) of the relevant coordinates of the system is derived. We show that for the improved Boussinesq (IBq) equation the set of ODEs has limiting cases leading to a set of ODEs which can be directly derived either from the ill-posed Boussinesq equation or from the Korteweg-de Vries (KdV) equation. The case of a soliton propagating in the presence of damping and thermal noise is considered for the IBq equation. A good agreement between theory and simulations is observed showing the strong robustness of these excitations. The results obtained here generalize previous results obtained in the frame of the KdV equation for lattice solitons in the monatomic chain of atoms.
Nonlinear polarization waves in a two-component Bose-Einstein condensate
NASA Astrophysics Data System (ADS)
Kamchatnov, A. M.; Kartashov, Y. V.; Larré, P.-É.; Pavloff, N.
2014-03-01
A two-component Bose-Einstein condensate whose dynamics is described by a system of coupled Gross-Pitaevskii equations accommodates waves with different symmetries. A first type of waves corresponds to excitations for which the motion of both components is locally in phase. For the second type of waves, the two components have a counterphase local motion. When the values of the inter- and intracomponent interaction constants are different, the long-wavelength behavior of these two modes corresponds to two types of sound with different velocities. In the limit of weak nonlinearity and small dispersion, the first mode is described by the well-known Korteweg-de Vries equation. In the same limit, we show that the second mode can be described by the Gardner equation if the values of the two intracomponent interaction constants are sufficiently close. This leads to a rich variety of nonlinear excitations (solitons, kinks, algebraic solitons, breathers) which do not exist in the Korteweg-de Vries description.
Dust acoustic solitary and shock excitations in a Thomas-Fermi magnetoplasma
Rahim, Z.; Qamar, A.; Ali, S.
2014-07-15
The linear and nonlinear properties of dust-acoustic waves are investigated in a collisionless Thomas-Fermi magnetoplasma, whose constituents are electrons, ions, and negatively charged dust particles. At dust time scale, the electron and ion number densities follow the Thomas-Fermi distribution, whereas the dust component is described by the classical fluid equations. A linear dispersion relation is analyzed to show that the wave frequencies associated with the upper and lower modes are enhanced with the variation of dust concentration. The effect of the latter is seen more strongly on the upper mode as compared to the lower mode. For nonlinear analysis, we obtain magnetized Korteweg-de Vries (KdV) and Zakharov-Kuznetsov (ZK) equations involving the dust-acoustic solitary waves in the framework of reductive perturbation technique. Furthermore, the shock wave excitations are also studied by allowing dissipation effects in the model, leading to the Korteweg-de Vries-Burgers (KdVB) and ZKB equations. The analysis reveals that the dust-acoustic solitary and shock excitations in a Thomas-Fermi plasma are strongly influenced by the plasma parameters, e.g., dust concentration, dust temperature, obliqueness, magnetic field strength, and dust fluid viscosity. The present results should be important for understanding the solitary and shock excitations in the environments of white dwarfs or supernova, where dust particles can exist.
The propagation of internal undular bores over variable topography
NASA Astrophysics Data System (ADS)
Grimshaw, R.; Yuan, C.
2016-10-01
In the coastal ocean, large amplitude, horizontally propagating internal wave trains are commonly observed. These are long nonlinear waves and can be modelled by equations of the Korteweg-de Vries type. Typically they occur in regions of variable bottom topography when the variable-coefficient Korteweg-de Vries equation is an appropriate model. Of special interest is the situation when the coefficient of the quadratic nonlinear term changes sign at a certain critical point. This case has been widely studied for a solitary wave, which is extinguished at the critical point and replaced by a train of solitary waves of the opposite polarity to the incident wave, riding on a pedestal of the original polarity. Here we examine the same situation for an undular bore, represented by a modulated periodic wave train. Numerical simulations and some asymptotic analysis based on Whitham modulation equations show that the leading solitary waves in the undular bore are destroyed and replaced by a developing rarefaction wave supporting emerging solitary waves of the opposite polarity. In contrast the rear of the undular bore emerges with the same shape, but with reduced wave amplitudes, a shorter overall length scale and moves more slowly.
Han, Qiang
2010-01-27
In this paper, we present a method to construct the eigenspace of the tight-binding electrons moving on a 2D square lattice with nearest-neighbor hopping in the presence of a perpendicular uniform magnetic field which imposes (quasi-)periodic boundary conditions for the wavefunctions in the magnetic unit cell. Exact unitary transformations are put forward to correlate the discrete eigenvectors of the 2D electrons with those of the Harper equation. The cyclic tridiagonal matrix associated with the Harper equation is then tridiagonalized by another unitary transformation. The obtained truncated eigenbasis is utilized to expand the Bogoliubov-de Gennes equations for the superconducting vortex lattice state, which shows the merit of our method in studying large-sized systems. To test our method, we have applied our results to study the vortex lattice state of an s-wave superconductor.
Averaging and renormalization for the Korteveg–deVries–Burgers equation
Chorin, Alexandre J.
2003-01-01
We consider traveling wave solutions of the Korteveg–deVries–Burgers equation and set up an analogy between the spatial averaging of these traveling waves and real-space renormalization for Hamiltonian systems. The result is an effective equation that reproduces means of the unaveraged, highly oscillatory, solution. The averaging enhances the apparent diffusion, creating an “eddy” (or renormalized) diffusion coefficient; the relation between the eddy diffusion coefficient and the original diffusion coefficient is found numerically to be one of incomplete similarity, setting up an instance of Barenblatt's renormalization group. The results suggest a relation between self-similar solutions of differential equations on one hand and renormalization groups and optimal prediction algorithms on the other. An analogy with hydrodynamics is pointed out. PMID:12913126
Wheeler-DeWitt equation and Lie symmetries in Bianchi scalar-field cosmology
NASA Astrophysics Data System (ADS)
Paliathanasis, A.; Karpathopoulos, L.; Wojnar, A.; Capozziello, S.
2016-04-01
Lie symmetries are discussed for the Wheeler-De Witt equation in Bianchi Class A cosmologies. In particular, we consider general relativity, minimally coupled scalar-field gravity and hybrid gravity as paradigmatic examples of the approach. Several invariant solutions are determined and classified according to the form of the scalar-field potential. The approach gives rise to a suitable method to select classical solutions and it is based on the first principle of the existence of symmetries.
Kanai, Masahiro; Isojima, Shin; Nishinari, Katsuhiro; Tokihiro, Tetsuji
2009-05-01
In this paper, we propose the ultradiscrete optimal velocity model, a cellular-automaton model for traffic flow, by applying the ultradiscrete method for the optimal velocity model. The optimal velocity model, defined by a differential equation, is one of the most important models; in particular, it successfully reproduces the instability of high-flux traffic. It is often pointed out that there is a close relation between the optimal velocity model and the modified Korteweg-de Vries (mkdV) equation, a soliton equation. Meanwhile, the ultradiscrete method enables one to reduce soliton equations to cellular automata which inherit the solitonic nature, such as an infinite number of conservation laws, and soliton solutions. We find that the theory of soliton equations is available for generic differential equations and the simulation results reveal that the model obtained reproduces both absolutely unstable and convectively unstable flows as well as the optimal velocity model.
NASA Astrophysics Data System (ADS)
Kanai, Masahiro; Isojima, Shin; Nishinari, Katsuhiro; Tokihiro, Tetsuji
2009-05-01
In this paper, we propose the ultradiscrete optimal velocity model, a cellular-automaton model for traffic flow, by applying the ultradiscrete method for the optimal velocity model. The optimal velocity model, defined by a differential equation, is one of the most important models; in particular, it successfully reproduces the instability of high-flux traffic. It is often pointed out that there is a close relation between the optimal velocity model and the modified Korteweg-de Vries (mkdV) equation, a soliton equation. Meanwhile, the ultradiscrete method enables one to reduce soliton equations to cellular automata which inherit the solitonic nature, such as an infinite number of conservation laws, and soliton solutions. We find that the theory of soliton equations is available for generic differential equations and the simulation results reveal that the model obtained reproduces both absolutely unstable and convectively unstable flows as well as the optimal velocity model.
NASA Astrophysics Data System (ADS)
Irfan, M.; Ali, S.; Mirza, Arshad M.
2016-02-01
Two-fluid quantum magnetohydrodynamic (QMHD) equations are employed to investigate linear and nonlinear properties of the magnetosonic waves in a semi-relativistic dense plasma accounting for degenerate relativistic electrons. In the linear analysis, a plane wave solution is used to derive the dispersion relation of magnetosonic waves, which is significantly modified due to relativistic degenerate electrons. However, for a nonlinear investigation of solitary and shock waves, we employ the reductive perturbation technique for the derivation of Korteweg-de Vries (KdV) and Korteweg-de Vries Burger (KdVB) equations, admitting nonlinear wave solutions. Numerically, it is shown that the wave frequency decreases to attain a lowest possible value at a certain critical number density Nc(0), and then increases beyond Nc(0) as the plasma number density increases. Moreover, the relativistic electrons and associated pressure degeneracy lead to a reduction in the spatial extents of the magnetosonic waves and a strengthening of the shock amplitude. The results might be important for understanding the linear and nonlinear magnetosonic excitations in dense astrophysical plasmas, such as in white dwarfs, magnetars and neutron stars, etc., where relativistic degenerate electrons are present.
Solutions to Yang-Mills Equations on Four-Dimensional de Sitter Space
NASA Astrophysics Data System (ADS)
Ivanova, Tatiana A.; Lechtenfeld, Olaf; Popov, Alexander D.
2017-08-01
We consider pure SU(2) Yang-Mills theory on four-dimensional de Sitter space dS4 and construct a smooth and spatially homogeneous magnetic solution to the Yang-Mills equations. Slicing dS4 as R ×S3, via an SU(2)-equivariant ansatz, we reduce the Yang-Mills equations to ordinary matrix differential equations and further to Newtonian dynamics in a double-well potential. Its local maximum yields a Yang-Mills solution whose color-magnetic field at time τ ∈R is given by B˜a=-1/2 Ia/(R2cosh2τ ), where Ia for a =1 , 2, 3 are the SU(2) generators and R is the de Sitter radius. At any moment, this spatially homogeneous configuration has finite energy, but its action is also finite and of the value -1/2 j (j +1 )(2 j +1 )π3 in a spin-j representation. Similarly, the double-well bounce produces a family of homogeneous finite-action electric-magnetic solutions with the same energy. There is a continuum of other solutions whose energy and action extend down to zero.
The Solution of Dirac Equation in Quasi-Extreme REISSNER-NORDSTRÖM de Sitter Space
NASA Astrophysics Data System (ADS)
Lyu, Yan; Cui, Song; Liu, Ling
The radial parts of Dirac equation between the outer black hole horizon and the cosmological horizon in quasi-extreme Reissner-Nordström de Sitter (RNdS) geometry is solved numerically. We use an accurate polynomial approximation to mimic the modified tortoise coordinate hat r*(r), for obtaining the inverse function r=r(hat r*) and V=V(hat r*). We then use a quantum mechanical method to solve the wave equation and give the reflection and transmission coefficients. We concentrate on two limiting cases. The first case is when the two horizons are close to each other, and the second case is when the horizons are far apart.
Numerical solution of the Dirac equation in Schwarzschild de Sitter spacetime
NASA Astrophysics Data System (ADS)
Lyu, Y.; Gui, Y. X.
2007-02-01
The radial parts of the Dirac equation between the inner and the outer horizon in Schwarzschild-de Sitter geometry are solved. Two limiting cases are concerned. The first case is when the two horizons are far apart and the second case is when the horizons are close to each other. In each case, a 'tangent' approximation is used to replace the modified 'tortoise' coordinate r*, which leads to a simple analytically invertible relation between r* and the radius r. The potential V(r*) is replaced by a collection of step functions in sequence. Then the solutions of the wave equation as well as the reflection and transmission coefficients are computed by a quantum mechanical method.
Huygens' principle for the Klein-Gordon equation in the de Sitter spacetime
Yagdjian, Karen
2013-09-15
In this article we prove that the Klein-Gordon equation in the de Sitter spacetime obeys the Huygens' principle only if the physical mass m of the scalar field and the dimension n⩾ 2 of the spatial variable are tied by the equation m{sup 2}= (n{sup 2}−1)/4. Moreover, we define the incomplete Huygens' principle, which is the Huygens' principle restricted to the vanishing second initial datum, and then reveals that the massless scalar field in the de Sitter spacetime obeys the incomplete Huygens' principle and does not obey the Huygens' principle, for the dimensions n= 1, 3, only. Thus, in the de Sitter spacetime the existence of two different scalar fields (in fact, with m= 0 and m{sup 2}= (n{sup 2}−1)/4), which obey incomplete Huygens' principle, is equivalent to the condition n= 3, the spatial dimension of the physical world. In fact, Paul Ehrenfest in 1917 addressed the question: “Why has our space just three dimensions?”. For n= 3 these two values of the mass are the endpoints of the so-called in quantum field theory the Higuchi bound. The value m{sup 2}= (n{sup 2}−1)/4 of the physical mass allows us also to obtain complete asymptotic expansion of the solution for the large time.
NASA Astrophysics Data System (ADS)
Hafez, M. G.; Talukder, M. R.; Ali, M. Hossain
2016-01-01
The theoretical and numerical studies have been investigated on the nonlinear propagation of electrostatic ion-acoustic waves (IAWs) in an un-magnetized Thomas-Fermi plasma system consisting of electron, positrons, and positive ions for both of ultra-relativistic and non-relativistic degenerate electrons. Korteweg-de Vries (K-dV) equation is derived from the model equations by using the well-known reductive perturbation method. This equation is solved by employing the generalized Riccati equation mapping method. The hyperbolic functions type solutions to the K-dV equation are only considered for describing the effect of plasma parameters on the propagation of electrostatic IAWs for both of ultra-relativistic and non-relativistic degenerate electrons. The obtained results may be helpful in proper understanding the features of small but finite amplitude localized IAWs in degenerate plasmas and provide the mathematical foundation in plasma physics.
The equations of relative motion in the orbital reference frame
NASA Astrophysics Data System (ADS)
Casotto, Stefano
2016-03-01
The analysis of relative motion of two spacecraft in Earth-bound orbits is usually carried out on the basis of simplifying assumptions. In particular, the reference spacecraft is assumed to follow a circular orbit, in which case the equations of relative motion are governed by the well-known Hill-Clohessy-Wiltshire equations. Circular motion is not, however, a solution when the Earth's flattening is accounted for, except for equatorial orbits, where in any case the acceleration term is not Newtonian. Several attempts have been made to account for the J_2 effects, either by ingeniously taking advantage of their differential effects, or by cleverly introducing ad-hoc terms in the equations of motion on the basis of geometrical analysis of the J_2 perturbing effects. Analysis of relative motion about an unperturbed elliptical orbit is the next step in complexity. Relative motion about a J_2-perturbed elliptic reference trajectory is clearly a challenging problem, which has received little attention. All these problems are based on either the Hill-Clohessy-Wiltshire equations for circular reference motion, or the de Vries/Tschauner-Hempel equations for elliptical reference motion, which are both approximate versions of the exact equations of relative motion. The main difference between the exact and approximate forms of these equations consists in the expression for the angular velocity and the angular acceleration of the rotating reference frame with respect to an inertial reference frame. The rotating reference frame is invariably taken as the local orbital frame, i.e., the RTN frame generated by the radial, the transverse, and the normal directions along the primary spacecraft orbit. Some authors have tried to account for the non-constant nature of the angular velocity vector, but have limited their correction to a mean motion value consistent with the J_2 perturbation terms. However, the angular velocity vector is also affected in direction, which causes precession
Exact solutions of the Wheeler-DeWitt equation and the Yamabe construction
NASA Astrophysics Data System (ADS)
Ita, Eyo Eyo, III; Soo, Chopin
2015-08-01
Exact solutions of the Wheeler-DeWitt equation of the full theory of four dimensional gravity of Lorentzian signature are obtained. They are characterized by Schrödinger wavefunctionals having support on 3-metrics of constant spatial scalar curvature, and thus contain two full physical field degrees of freedom in accordance with the Yamabe construction. These solutions are moreover Gaussians of minimum uncertainty and they are naturally associated with a rigged Hilbert space. In addition, in the limit the regulator is removed, exact 3-dimensional diffeomorphism and local gauge invariance of the solutions are recovered.
Exact solutions of the Wheeler–DeWitt equation and the Yamabe construction
Ita III, Eyo Eyo; Soo, Chopin
2015-08-15
Exact solutions of the Wheeler–DeWitt equation of the full theory of four dimensional gravity of Lorentzian signature are obtained. They are characterized by Schrödinger wavefunctionals having support on 3-metrics of constant spatial scalar curvature, and thus contain two full physical field degrees of freedom in accordance with the Yamabe construction. These solutions are moreover Gaussians of minimum uncertainty and they are naturally associated with a rigged Hilbert space. In addition, in the limit the regulator is removed, exact 3-dimensional diffeomorphism and local gauge invariance of the solutions are recovered.
Analytical solution of the geodesic equation in Kerr-(anti-) de Sitter space-times
Hackmann, Eva; Laemmerzahl, Claus; Kagramanova, Valeria; Kunz, Jutta
2010-02-15
The complete analytical solutions of the geodesic equations in Kerr-de Sitter and Kerr-anti-de Sitter space-times are presented. They are expressed in terms of Weierstrass elliptic p, {zeta}, and {sigma} functions as well as hyperelliptic Kleinian {sigma} functions restricted to the one-dimensional {theta} divisor. We analyze the dependency of timelike geodesics on the parameters of the space-time metric and the test-particle and compare the results with the situation in Kerr space-time with vanishing cosmological constant. Furthermore, we systematically can find all last stable spherical and circular orbits and derive the expressions of the deflection angle of flyby orbits, the orbital frequencies of bound orbits, the periastron shift, and the Lense-Thirring effect.
NASA Astrophysics Data System (ADS)
Hainzl, Christian; Seyrich, Jonathan
2016-05-01
In this paper we report on the results of a numerical study of the nonlinear time-dependent Bardeen-Cooper-Schrieffer (BCS) equations, often also denoted as Bogoliubov-de-Gennes (BdG) equations, for a one-dimensional system of fermions with contact interaction. We show that, even above the critical temperature, the full equations and their linear approximation give rise to completely different evolutions. In contrast to its linearization, the full nonlinear equation does not show any diffusive behavior in the order parameter. This means that the order parameter does not follow a Ginzburg-Landau-type of equation, in accordance with a recent theoretical result in [R.L. Frank, C. Hainzl, B. Schlein, R. Seiringer, to appear in Lett. Math. Phys., arXiv:1504.05885 (2016)]. We include a full description on the numerical implementation of the partial differential BCS/BdG equations.
NASA Astrophysics Data System (ADS)
Karmakar, P. K.; Borah, B.
2013-09-01
We try to present a theoretical evolutionary model leading to the excitations of nonlinear pulsational eigenmodes in a planar (1D) collisional dust molecular cloud (DMC) on the Jeans scale. The basis of the adopted model is the Jeans assumption of self-gravitating homogeneous uniform medium for simplification. It is a self-gravitating multi-fluid consisting of the Boltzmann distributed warm electrons and ions, and the inertial cold dust grains with partial ionization. Dust-charge fluctuations, convections and all the possible collisions are included. The grain-charge behaves as a dynamical variable owing mainly to the attachment of the electrons and ions to the grain-surfaces randomly. The adopted technique is centered around a mathematical model based on new solitary spectral patterns within the hydrodynamic framework. The collective dynamics of the patterns is governed by driven Korteweg-de Vries ( d-KdV) and Korteweg-de Vries (KdV) equations obtained by a standard multiscale analysis. Then, simplified analytical and numerical solutions are presented. The grain-charge fluctuation and collision processes play a key role in the DMC stability. The sensitive dependence of the eigenmode amplitudes on diverse relevant plasma parameters is discussed. The significance of the main results in astrophysical, laboratory and space environments are concisely summarized.
Solitary wave dynamics in shallow water over periodic topography.
Nakoulima, Ousseynou; Zahibo, Narcisse; Pelinovsky, Efim; Talipova, Tatiana; Kurkin, Andrey
2005-09-01
The problem of long-wave scattering by piecewise-constant periodic topography is studied both for a linear solitary-like wave pulse, and for a weakly nonlinear solitary wave [Korteweg-de Vries (KdV) soliton]. If the characteristic length of the topographic irregularities is larger than the pulse length, the solution of the scattering problem is obtained analytically for a leading wave in the framework of linear shallow-water theory. The wave decrement in the case of the small height of the topographic irregularities is proportional to delta2, where delta is the relative height of the topographic obstacles. An analytical approximate solution is also obtained for the weakly nonlinear problem when the length of the irregularities is larger than the characteristic nonlinear length scale. In this case, the Korteweg-de Vries equation is solved for each piece of constant depth by using the inverse scattering technique; the solutions are matched at each step by using linear shallow-water theory. The weakly nonlinear solitary wave decays more significantly than the linear solitary pulse. Solitary wave dynamics above a random seabed is also discussed, and the results obtained for random topography (including experimental data) are in reasonable agreement with the calculations for piecewise topography.
Singh, Dhananjay K.; Malik, Hitendra K.
2007-11-15
Considering an inhomogeneous plasma having finite-temperature negative and positive ions, and the isothermal electrons in the presence of an external magnetic field, the solitons at noncritical and critical densities of the negative ions are studied through Korteweg-deVries (KdV) and modified Korteweg-deVries (mKdV) equations, respectively. The compressive (rarefactive) KdV solitons are found to propagate when the negative ion concentration is less (greater) than the critical density of the negative ions. At the critical density, both the compressive and the rarefactive solitons of equal amplitudes are found to occur. The energies of the compressive KdV soliton and the mKdV solitons are found to increase and that of the rarefactive KdV soliton is found to decrease with the negative ion density. Soliton energy for both the KdV and the mKdV solitons gets lowered under the effect of stronger magnetic field. The effect of ion temperature is to increase the energy of the compressive KdV soliton, whereas the energy of the rarefactive KdV soliton as well as of the mKdV solitons gets decreased. The variation of the energy with the obliqueness of the magnetic field is different for the KdV and the mKdV solitons.
Soliton turbulence in shallow water ocean surface waves.
Costa, Andrea; Osborne, Alfred R; Resio, Donald T; Alessio, Silvia; Chrivì, Elisabetta; Saggese, Enrica; Bellomo, Katinka; Long, Chuck E
2014-09-05
We analyze shallow water wind waves in Currituck Sound, North Carolina and experimentally confirm, for the first time, the presence of soliton turbulence in ocean waves. Soliton turbulence is an exotic form of nonlinear wave motion where low frequency energy may also be viewed as a dense soliton gas, described theoretically by the soliton limit of the Korteweg-deVries equation, a completely integrable soliton system: Hence the phrase "soliton turbulence" is synonymous with "integrable soliton turbulence." For periodic-quasiperiodic boundary conditions the ergodic solutions of Korteweg-deVries are exactly solvable by finite gap theory (FGT), the basis of our data analysis. We find that large amplitude measured wave trains near the energetic peak of a storm have low frequency power spectra that behave as ∼ω-1. We use the linear Fourier transform to estimate this power law from the power spectrum and to filter densely packed soliton wave trains from the data. We apply FGT to determine the soliton spectrum and find that the low frequency ∼ω-1 region is soliton dominated. The solitons have random FGT phases, a soliton random phase approximation, which supports our interpretation of the data as soliton turbulence. From the probability density of the solitons we are able to demonstrate that the solitons are dense in time and highly non-Gaussian.
NASA Astrophysics Data System (ADS)
Cao, Jin-Liang; Shi, Zhong-Ke
2016-04-01
In this paper, a novel hydrodynamic lattice model is proposed by considering of relative current for two-lane gradient road system. The stability condition is obtained by using linear stability theory and shown that the stability of traffic flow varies with three parameters, that is, the slope, the sensitivity of response to the relative current and the rate of lane changing. The stable region increases with the increasing of one of them when another two parameters are constant. By using nonlinear analysis, the Burgers, Korteweg-de Vries, and modified Korteweg-de Vries equations are derived to describe the phase transition of traffic flow. Their solutions present the density wave as the triangular shock wave, soliton wave, and kink-antikink wave in the stable, metastable, and unstable region, respectively, which can explain the phase transitions from free traffic to stop-and-go traffic, and finally to congested traffic. To verify the theoretical results, a series of numerical simulations are carried out. The numerical results are consistent with the analytical results. To check the novel model, calibration are taken based on the empirical traffic flow data. The theoretical results and numerical results show that the traffic flow on the gradient road becomes more stable and the traffic congestion can be efficiently suppressed by considering the relative current and lane changing, and the empirical analysis shows that the novel lattice model is reasonable.
Nonlinear Electromagnetic Waves in a Degenerate Electron-Positron Plasma
NASA Astrophysics Data System (ADS)
El-Labany, S. K.; El-Taibany, W. F.; El-Samahy, A. E.; Hafez, A. M.; Atteya, A.
2015-08-01
Using the reductive perturbation technique (RPT), the nonlinear propagation of magnetosonic solitary waves in an ultracold, degenerate (extremely dense) electron-positron (EP) plasma (containing ultracold, degenerate electron, and positron fluids) is investigated. The set of basic equations is reduced to a Korteweg-de Vries (KdV) equation for the lowest-order perturbed magnetic field and to a KdV type equation for the higher-order perturbed magnetic field. The solutions of these evolution equations are obtained. For better accuracy and searching on new features, the new solutions are analyzed numerically based on compact objects (white dwarf) parameters. It is found that including the higher-order corrections results as a reduction (increment) of the fast (slow) electromagnetic wave amplitude but the wave width is increased in both cases. The ranges where the RPT can describe adequately the total magnetic field including different conditions are discussed.
NASA Astrophysics Data System (ADS)
Dimitrova, Zlatinka I.
2015-12-01
We investigate flow of incompressible fluid in a cylindrical tube with elastic walls. The radius of the tube may change along its length. The discussed problem is connected to the fluid-structure interaction in large human arteries and especially to nonlinear effects. The long-wave approximation is applied to solve model equations. The obtained model Korteweg-deVries equation possessing a variable coefficient is reduced to a nonlinear dynamical system of three first order differential equations. The low probability of a solitary wave arising is shown. Periodic wave solutions of the model system of equations are studied and it is shown that the waves, that are consequence of the irregular heart pulsations may be modelled by a sequence of parts of such periodic wave solutions.
Kinetic treatment of nonlinear ion-acoustic waves in multi-ion plasma
NASA Astrophysics Data System (ADS)
Ahmad, Zulfiqar; Ahmad, Mushtaq; Qamar, A.
2017-09-01
By applying the kinetic theory of the Valsove-Poisson model and the reductive perturbation technique, a Korteweg-de Vries (KdV) equation is derived for small but finite amplitude ion acoustic waves in multi-ion plasma composed of positive and negative ions along with the fraction of electrons. A correspondent equation is also derived from the basic set of fluid equations of adiabatic ions and isothermal electrons. Both kinetic and fluid KdV equations are stationary solved with different nature of coefficients. Their differences are discussed both analytically and numerically. The criteria of the fluid approach as a limiting case of kinetic theory are also discussed. The presence of negative ion makes some modification in the solitary structure that has also been discussed with its implication at the laboratory level.
Gardner's deformation of the Krasil'shchik—Kersten system
NASA Astrophysics Data System (ADS)
Kiselev, Arthemy V.; Krutov, Andrey O.
2015-06-01
The classical problem of construction of Gardner's deformations for infinite-dimensional completely integrable systems of evolutionary partial differential equations (PDE) amounts essentially to finding the recurrence relations between the integrals of motion. Using the correspondence between the zero-curvature representations and Gardner deformations for PDE, we construct a Gardner's deformation for the Krasil'shchik-Kersten system. For this, we introduce the new nonlocal variables in such a way that the rules to differentiate them are consistent by virtue of the equations at hand and second, the full system of Krasil'shchik-Kersten's equations and the new rules contains the Korteweg-de Vries equation and classical Gardner's deformation for it.
NASA Astrophysics Data System (ADS)
Paul, S. N.; Chatterjee, A.; Paul, Indrani
2017-01-01
Nonlinear propagation of ion-acoustic waves in self-gravitating multicomponent dusty plasma consisting of positive ions, non-isothermal two-temperature electrons and negatively charged dust particles with fluctuating charges and drifting ions has been studied using the reductive perturbation method. It has been shown that nonlinear propagation of ion-acoustic waves in gravitating dusty plasma is described by an uncoupled third order partial differential equation which is a modified form of Korteweg-deVries equation, in contraries to the coupled nonlinear equations obtained by earlier authors. Quasi-soliton solution for the ion-acoustic solitary wave has been obtained from this uncoupled nonlinear equation. Effects of non-isothermal two-temperature electrons, gravity, dust charge fluctuation and drift motion of ions on the ion-acoustic solitary waves have been discussed.
Dust acoustic dressed soliton with dust charge fluctuations
Asgari, H.; Muniandy, S. V.; Wong, C. S.
2010-06-15
Modeling of dust acoustic solitons observed in dusty plasma experiment [Bandyopadhyay et al., Phys. Rev. Lett. 101, 065006 (2008)] using the Korteweg-de Vries (KdV) equation showed significant discrepancies in the regime of large amplitudes (or high soliton speed). In this paper, higher order perturbation corrections to the standard KdV soliton are proposed and the resulting dressed soliton is shown to describe the experimental data better, in particular, at high soliton speed. The effects of dust charge fluctuations on the dust acoustic dressed soliton in a dusty plasma system are also investigated. The KdV equation and a linear inhomogeneous equation, governing the evolution of first and second order potentials, respectively, are derived for the system by using reductive perturbation technique. Renormalization procedure is used to obtain nonsecular solutions of these coupled equations. The characteristics of dust acoustic dressed solitons with and without dust charge fluctuations are discussed.
Higher-order corrections to dust ion-acoustic soliton in a quantum dusty plasma
Chatterjee, Prasanta; Das, Brindaban; Mondal, Ganesh; Muniandy, S. V.; Wong, C. S.
2010-10-15
Dust ion-acoustic soliton is studied in an electron-dust-ion plasma by employing a two-fluid quantum hydrodynamic model. Ions and electrons are assumed to follow quantum mechanical behaviors in dust background. The Korteweg-de Vries (KdV) equation and higher order contribution to KdV equations are derived using reductive perturbation technique. The higher order contribution is obtained as a higher order inhomogeneous differential equation. The nonsecular solution of the higher order contribution is obtained by using the renormalization method and the particular solution of the inhomogeneous equation is determined using a truncated series solution method. The effects of dust concentration, quantum parameter for ions and electrons, and soliton velocity on the amplitude and width of the dressed soliton are discussed.
Head on collision of multi-solitons in an electron-positron-ion plasma having superthermal electrons
Roy, Kaushik; Chatterjee, Prasanta Roychoudhury, Rajkumar
2014-10-15
The head-on collision and overtaking collision of four solitons in a plasma comprising superthermal electrons, cold ions, and Boltzmann distributed positrons are investigated using the extended Poincare-Lighthill-Kuo (PLK) together with Hirota's method. PLK method yields two separate Korteweg-de Vries (KdV) equations where solitons obtained from any KdV equation move along a direction opposite to that of solitons obtained from the other KdV equation, While Hirota's method gives multi-soliton solution for each KdV equation all of which move along the same direction where the fastest moving soliton eventually overtakes the other ones. We have considered here two soliton solutions obtained from Hirota's method. Phase shifts acquired by each soliton due to both head-on collision and overtaking collision are calculated analytically.
A numerical study of nonlinear waves in a transcritical flow of stratified fluid past an obstacle
NASA Astrophysics Data System (ADS)
Hanazaki, Hideshi
1992-10-01
A numerical study of the flow of stratified fluid past an obstacle in a horizontal channel is described. Upstream advancing of waves near critically (resonance) appears in the case of ordinary two-layer flow, in which case the flow is described well by the solution of the forced extended Korteweg-de Vries (KdV) equation which has a cubic nonlinear term. It is shown theoretically that the upstream waves in the general two-layer flow cannot be well described by the forced KdV equation except when the wave amplitude is very small. The critical-level flow is also governed by the forced extended KdV equation. However, because of the smallness of the coefficient of the quadratic nonlinear term, the bore cannot propagate upstream at exact resonance. The results for the linearly stratified Boussinesq flow show good agreement with the solution of the Grimshaw and Yi (1991) equation, at least for exact resonance.
NASA Astrophysics Data System (ADS)
Blyakhman, L. G.; Gromov, E. M.; Onosova, I. V.; Tyutin, V. V.
2017-05-01
The dynamics of a two-component Davydov-Scott (DS) soliton with a small mismatch of the initial location or velocity of the high-frequency (HF) component was investigated within the framework of the Zakharov-type system of two coupled equations for the HF and low-frequency (LF) fields. In this system, the HF field is described by the linear Schrödinger equation with the potential generated by the LF component varying in time and space. The LF component in this system is described by the Korteweg-de Vries equation with a term of quadratic influence of the HF field on the LF field. The frequency of the DS soliton's component oscillation was found analytically using the balance equation. The perturbed DS soliton was shown to be stable. The analytical results were confirmed by numerical simulations.
NASA Astrophysics Data System (ADS)
Masood, W.; Faryal, Anam; Siddiq, M.
2017-10-01
The propagation of one dimensional nonlinear electrostatic waves in unmagnetized pair-ion-electron (PIE) plasmas comprising of oppositely charged inertial ions of equal mass but different temperatures and Boltzmann electrons is investigated. In the linear analysis, the acquired biquadratic dispersion relation yields fast and slow modes for PIE plasmas. In the nonlinear regime, the Gardner equation in PIE plasmas is derived in the weak nonlinearity limit. The plasma parameter regime is explicitly shown where the Korteweg de Vries equation used in the earlier studies is no longer valid and the Gardner equation becomes relevant. Solitary and kink solutions of Gardner equation are also presented. Interestingly, it has been observed that these solutions exist for the fast mode; however, no such structure is found to exist for the slow mode. It is hoped that the present study would be beneficial to understand the solitary and kink solutions in laboratory produced PIE plasmas and parametric regimes in which this study is applicable.
NASA Astrophysics Data System (ADS)
Hafez, M. G.; Talukder, M. R.; Hossain Ali, M.
2016-01-01
The Korteweg-de Vries Burgers (KdVB) -like equation is derived to study the characteristics of nonlinear propagation of ion acoustic solitions in a highly relativistic plasma containing relativistic ions and nonextensive distribution of electrons and positrons using the well known reductive perturbation technique. The KdVB-like equation is solved employing the Bernoulli's equation method taking unperturbed positron to electron concentration ratio, electron to positron temperature ratio, strength of nonextensivity, ion kinematic viscosity, and highly relativistic streaming factor. It is found that these parameters significantly modify the structures of the solitonic excitation. The ion acoustic shock profiles are observed due to the influence of ion kinematic viscosity. In the absence of dissipative term to the KdVB equation, compressive and rarefactive solitons are observed in case of superthermality, but only compressive solitons are found for the case of subthermality.
From Nothing to Something II: Nonlinear Systems via Consistent Correlated Bang
NASA Astrophysics Data System (ADS)
Lou, Sen-Yue
2017-06-01
Chinese ancient sage Laozi said everything comes from \\emph{\\bf \\em "nothing"}. \\rm In the first letter (Chin. Phys. Lett. 30 (2013) 080202), infinitely many discrete integrable systems have been obtained from "nothing" via simple principles (Dao). In this second letter, a new idea, the consistent correlated bang, is introduced to obtain nonlinear dynamic systems including some integrable ones such as the continuous nonlinear Schr\\"odinger equation (NLS), the (potential) Korteweg de Vries (KdV) equation, the (potential) Kadomtsev-Petviashvili (KP) equation and the sine-Gordon (sG) equation. These nonlinear systems are derived from nothing via suitable "Dao", the shifted parity, the charge conjugate, the delayed time reversal, the shifted exchange, the shifted-parity-rotation and so on.
Application of the (G'/G)-expansion method to nonlinear blood flow in large vessels
NASA Astrophysics Data System (ADS)
Kol, Guy Richard; Bertrand Tabi, Conrad
2011-04-01
As is widely known today, Navier-Stokes equations are used to describe blood flow in large vessels. In the past several decades, and even in very recent works, these equations have been reduced to Korteweg-de Vries (KdV), modified KdV or Boussinesq equations. In this paper, we avoid such simplifications and investigate the analytical traveling wave solutions of the one-dimensional generic Navier-Stokes equations, through the (G ' /G)-expansion method. These traveling wave solutions include hyperbolic functions, trigonometric functions and rational functions. Since some of them are not yet explored in the study of blood flow, we pay attention to hyperbolic function solutions and we show that the (G ' /G)-expansion method presents a wider applicability that allows us to bring out the widely known blood flow behaviors. The biological implications of the found solutions are discussed accordingly.
VizieR Online Data Catalog: AQ Boo VRI differential light curves (Wang+, 2016)
NASA Astrophysics Data System (ADS)
Wang, S.; Zhang, L.; Pi, Q.; Han, X. L.; Zhang, X.; Lu, H.; Wang, D.; Li, T.
2016-11-01
On March 22 and April 19 in 2014, we observed AQ Boo with the 60cm telescope at Xinglong Station of the National Astronomical Observatories of China (NAOC). The CCD camera on this telescope has a resolution of 1024 x 1024 pixels and its corresponding field of view is 17'x17' (Yang, 2013NewA...25..109Y). The other three days of data were obtained using the 1-m telescope at Yunnan Observatory of Chinese Academy of Sciences, on January 20, 21 and February 28 in 2015. The CCD camera on this telescope has a resolution of 2048x2048 pixels and its corresponding field of view is 7.3'x7.3'. Bessel VRI filters were used. The exposure times are 100-170s, 50-100s and 50-80s in the V, R, I bands, respectively. (1 data file).
Some notes on the Gunn-Stryker spectrophotometry and synthetic VRI colors
NASA Astrophysics Data System (ADS)
Taylor, Benjamin J.; Joner, Michael D.
1990-09-01
Cousins VRI photometry is presented for 26 stars with continuous scans by Gunn and Stryker. This photometry is combined with literature data and a few unpublished results to critique synthetic colors from the Gunn-Stryker scans. For V - R, it is found that all pertinent results are consistent at the several-mmag level. For R - I, however, systematic differences are found which are most simply interpreted as a declination effect in the Gunn-Stryker scans. In addition, it is found that the Gunn-Stryker synthetic colors are unexpectedly noisy, with sigma per datum of about 0.02 mag. It is suggested that future users of the Gunn-Stryker data keep both these effects in mind.
Ion acoustic kinetic Alfvén rogue waves in two temperature electrons superthermal plasmas
NASA Astrophysics Data System (ADS)
Kaur, Nimardeep; Saini, N. S.
2016-10-01
The propagation properties of ion acoustic kinetic Alfvén (IAKA) solitary and rogue waves have been investigated in two temperature electrons magnetized superthermal plasma in the presence of dust impurity. A nonlinear analysis is carried out to derive the Korteweg-de Vries (KdV) equation using the reductive perturbation method (RPM) describing the evolution of solitary waves. The effect of various plasma parameters on the characteristics of the IAKA solitary waves is studied. The dynamics of ion acoustic kinetic Alfvén rogue waves (IAKARWs) are also studied by transforming the KdV equation into nonlinear Schrödinger (NLS) equation. The characteristics of rogue wave profile under the influence of various plasma parameters (κc, μc, σ , θ) are examined numerically by using the data of Saturn's magnetosphere (Schippers et al. 2008; Sakai et al. 2013).
Solitary and freak waves in superthermal plasma with ion jet
NASA Astrophysics Data System (ADS)
Abdelsalam, U. M.; Abdelsalam
2013-06-01
The nonlinear solitary and freak waves in a plasma composed of positive and negative ions, superthermal electrons, ion beam, and stationary dust particles have been investigated. The reductive perturbation method is used to obtain the Korteweg-de Vries (KdV) equation describing the system. The latter admits solitary wave solution, while the dynamics of the modulationally unstable wavepackets described by the KdV equation gives rise to the formation of freak/rogue excitation described by the nonlinear Schrödinger equation. In order to show that the characteristics of solitary and freak waves are influenced by plasma parameters, relevant numerical analysis of appropriate nonlinear solutions are presented. The results from this work predict nonlinear excitations that may associate with ion jet and superthermal electrons in Herbig-Haro objects.
NASA Astrophysics Data System (ADS)
Liu, Fangxun; Cheng, Rongjun; Ge, Hongxia; Yu, Chenyan
2016-12-01
In this study, a new car-following model is proposed based on taking the effect of the leading vehicle's velocity difference between the current speed and the historical speed into account. The model's linear stability condition is obtained via the linear stability theory. The time-dependent Ginzburg-Landau (TDGL) equation and the modified Korteweg-de Vries (mKdV) equation are deduced through the nonlinear analysis. The kink-antikink soliton can interpret the traffic jams near the critical point. In addition, the connection between the TDGL and the mKdV equations is also given. Numerical simulation shows that the new model can improve the stability of traffic flow, which is consistent with the theoretical analysis correspondingly.
Generation of zonal flows by coupled electrostatic drift and ion-acoustic waves
NASA Astrophysics Data System (ADS)
Kaladze, T. D.; Kahlon, L. Z.; Tsamalashvili, L. V.
2017-07-01
Generation of sheared zonal flow by low-frequency coupled electrostatic drift and ion-acoustic waves is presented. Primary waves of different (small, intermediate, and large) scales are considered, and the appropriate system of equations consisting of generalized Hasegawa-Mima equation for the electrostatic potential (involving both vector and scalar nonlinearities) and equation of parallel to magnetic field ions motion is obtained. It is shown that along with the mean poloidal flow with strong variation in minor radius mean sheared toroidal flow can also be generated. According to laboratory plasma experiments, main attention to large scale drift-ion-acoustic wave is given. Peculiarities of the Korteweg-de Vries type scalar nonlinearity due to the electrons temperature non-homogeneity in the formation of zonal flow by large-scale turbulence are widely discussed. Namely, it is observed that such type of flows need no generation condition and can be spontaneously excited.
NASA Astrophysics Data System (ADS)
Hussain, S.; Mahmood, S.
2017-10-01
Ion-acoustic shock wave propagation in dense magnetized plasmas with relative density effects of spin-up and spin-down degenerate electrons is studied. The ions are classical, and their dissipative effects on plasma dynamics are included via kinematic viscosity. The electrons with spin-up and spin-down states are taken as separate species. The quantum tunneling effects of electrons are also considered in equations of motions of electrons. The Korteweg de Vries Burgers (KdVB) equation is derived, which admits the shock solution. The KdVB equation is solved numerically to study the transition from shock with oscillatory trails at its wave fronts to the monotonic shock structure with respect to variations in different plasma parameters. The parametric role of the spin density polarization ratio in the propagation characteristics of the shock wave structure is discussed.
Rogue-wave bullets in a composite (2+1)D nonlinear medium.
Chen, Shihua; Soto-Crespo, Jose M; Baronio, Fabio; Grelu, Philippe; Mihalache, Dumitru
2016-07-11
We show that nonlinear wave packets localized in two dimensions with characteristic rogue wave profiles can propagate in a third dimension with significant stability. This unique behavior makes these waves analogous to light bullets, with the additional feature that they propagate on a finite background. Bulletlike rogue-wave singlet and triplet are derived analytically from a composite (2+1)D nonlinear wave equation. The latter can be interpreted as the combination of two integrable (1+1)D models expressed in different dimensions, namely, the Hirota equation and the complex modified Korteweg-de Vries equation. Numerical simulations confirm that the generation of rogue-wave bullets can be observed in the presence of spontaneous modulation instability activated by quantum noise.
Collisionless damping of dust-acoustic waves in a charge varying dusty plasma with nonextensive ions
Amour, Rabia; Tribeche, Mouloud
2014-12-15
The charge variation induced nonlinear dust-acoustic wave damping in a charge varying dusty plasma with nonextensive ions is considered. It is shown that the collisionless damping due to dust charge fluctuation causes the nonlinear dust acoustic wave propagation to be described by a damped Korteweg-de Vries (dK-dV) equation the coefficients of which depend sensitively on the nonextensive parameter q. The damping term, solely due to the dust charge variation, is affected by the ion nonextensivity. For the sake of completeness, the possible effects of nonextensivity and collisionless damping on weakly nonlinear wave packets described by the dK-dV equation are succinctly outlined by deriving a nonlinear Schrödinger-like equation with a complex nonlinear coefficient.
A formula relating sojourn times to the time of arrival in Hamiltonian dynamics
NASA Astrophysics Data System (ADS)
Gournay, A.; Tiedra de Aldecoa, R.
2012-06-01
We consider on a manifold M equipped with a Poisson bracket { ·, ·} a Hamiltonian H with complete flow and a family Φ ≡ (Φ1, …, Φd) of abstract position observables satisfying the condition {{Φj, H}, H} = 0 for each j. Under these assumptions, we prove a new formula relating sojourn times in dilated regions defined in terms of Φ to the time of arrival of classical orbits. The correspondence between this formula and a formula established recently in the framework of quantum mechanics is put into evidence. Among other examples, our theory applies to Stark Hamiltonians, homogeneous Hamiltonians, purely kinetic Hamiltonians, the repulsive harmonic potential, central force systems, the Poincaré ball model, the wave equation, the nonlinear Schrödinger equation, the Korteweg-de Vries equation and quantum Hamiltonians defined via expectation values.
Solitary and freak waves in a dusty plasma with negative ions
Abdelsalam, U. M.; Moslem, W. M.; Khater, A. H.; Shukla, P. K.
2011-09-15
It is shown that solitary and freak waves can propagate in a dusty plasma composed of positive and negative ions, as well as nonextensive electrons. The evolution of the solitary waves is described by the Korteweg-de Vries (KdV) equation. However, when the frequency of the carrier wave is much smaller than the ion plasma frequency then the KdV equation is also used to study the nonlinear evolution of modulationally unstable modified ion-acoustic wavepackets through the derivation of the nonlinear Schroedinger (NLS) equation. In order to show that the characteristics of the solitary and freak waves are influenced by the plasma parameters, the relevant numerical analysis of the appropriate nonlinear solutions is presented. The relevance of the present investigation to nonlinear waves in astrophysical plasma environments is discussed.
Nonlinear compressional waves in a two-dimensional Yukawa lattice.
Avinash, K; Zhu, P; Nosenko, V; Goree, J
2003-10-01
A modified Korteweg-de Vries (KdV) equation is obtained for studying the propagation of nonlinear compressional waves and pulses in a chain of particles including the effect of damping. Suitably altering the linear phase velocity makes this equation useful also for the problem of phonon propagation in a two-dimensional (2D) lattice. Assuming a Yukawa potential, we use this method to model compressional wave propagation in a 2D plasma crystal, as in a recent experiment. By integrating the modified KdV equation the pulse is allowed to evolve, and good agreement with the experiment is found. It is shown that the speed of a compressional pulse increases with its amplitude, while the speed of a rarefactive pulse decreases. It is further discussed how the drag due to the background gas has a crucial role in weakening nonlinear effects and preventing the emergence of a soliton.
Semiclassical solitons in strongly correlated systems of ultracold bosonic atoms in optical lattices
Demler, Eugene; Maltsev, Andrei
2011-07-15
Highlights: > Dynamics of their formation in strongly correlated systems of ultracold bosonic atoms in optical lattices. > Regime of very strong interactions between atoms, the so-called hard core bosons regime. > Character of soliton excitations is dramatically different from the usual Gross-Pitaevskii regime. - Abstract: We investigate theoretically soliton excitations and dynamics of their formation in strongly correlated systems of ultracold bosonic atoms in two and three dimensional optical lattices. We derive equations of nonlinear hydrodynamics in the regime of strong interactions and incommensurate fillings, when atoms can be treated as hard core bosons. When parameters change in one direction only we obtain Korteweg-de Vries type equation away from half-filling and modified KdV equation at half-filling. We apply this general analysis to a problem of the decay of the density step. We consider stability of one dimensional solutions to transverse fluctuations. Our results are also relevant for understanding nonequilibrium dynamics of lattice spin models.
Modulational instability of co-propagating internal wavetrains under rotation.
Whitfield, A J; Johnson, E R
2015-02-01
Weakly-nonlinear unidirectional long internal waves in a non-rotating frame are well described by the Korteweg-de Vries equation (KdV). Within the KdV framework, all isolated monochromatic wavetrains are stable to modulational instability. However, analysis of a coupled nonlinear Schrödinger equation system (CNLS) has shown that all systems of two co-propagating monochromatic wavetrains in the KdV are modulationally unstable. To take into account the effect of the background rotation of the Earth on long internal waves, this analysis is extended here to derive the CNLS for the rotation-modified KdV, or Ostrovsky, equation. Rotation stabilises wavetrain pairs when the wavelengths of both waves comprising the wavetrains are longer than the linear wave with maximum group velocity. The particular case when the wavetrains have different wavenumbers but the same linear group speed is emphasised.
Dressed ion-acoustic solitons in magnetized dusty plasmas
El-Labany, S. K.; El-Shamy, E. F.; El-Warraki, S. A.
2009-01-15
In the present research paper, the characteristics of ion acoustic solitary waves are investigated in hot magnetized dusty plasmas consisting of negatively charged dust grains, positively charged ion fluid, and isothermal electrons. Applying a reductive perturbation theory, a nonlinear Korteweg-de Vries (KdV) equation for the first-order perturbed potential and a linear inhomogeneous KdV-type equation for the second-order perturbed potentials are derived. Stationary solutions of these coupled equations are obtained using a renormalization method. The effects of the external oblique magnetic field, hot ion fluid, and higher-order nonlinearity on the nature of the ion acoustic solitary waves are discussed. The results complement and provide new insights into previously published results on this problem [R. S. Tiwari and M. K. Mishra, Phys. Plasmas 13, 062112 (2006)].
NASA Astrophysics Data System (ADS)
Cisneros-Ake, Luis A.; Solano Peláez, José F.
2017-05-01
The problem of energy transportation along a cubic anharmonic crystal lattice, in the unidirectional long wave limit, is considered. A detailed process, in the discrete lattice equations, shows that unidirectional stable propagating waves for the continuum limit produce a coupled system between a nonlinear Schrödinger (NLS) equation and the Korteweg-de Vries (KdV) equation. The traveling wave formalism provides a diversity of exact solutions ranging from the classical Davydov's soliton (subsonic and supersonic) of the first and second kind to a class consisting in the coupling between the KdV soliton and dark solitons containing the typical ones (similar to the dark-gray soliton in the standard defocusing NLS) and a new kind in the form of a two-hump dark soliton. This family of exact solutions are numerically tested, by means of the pseudo spectral method, in our NLS-KdV system.
Beyond the KdV: Post-explosion development.
Ostrovsky, L; Pelinovsky, E; Shrira, V; Stepanyants, Y
2015-09-01
Several threads of the last 25 years' developments in nonlinear wave theory that stem from the classical Korteweg-de Vries (KdV) equation are surveyed. The focus is on various generalizations of the KdV equation which include higher-order nonlinearity, large-scale dispersion, and a non-local integral dispersion. We also discuss how relatively simple models can capture strongly nonlinear dynamics and how various modifications of the KdV equation lead to qualitatively new, non-trivial solutions and regimes of evolution observable in the laboratory and in nature. As the main physical example, we choose internal gravity waves in the ocean for which all these models are applicable and have genuine importance. We also briefly outline the authors' view of the future development of the chosen lines of nonlinear wave theory.
Effect of a damping force on dust acoustic waves simulated by particle-in-cell method
NASA Astrophysics Data System (ADS)
Gao, Dong-Ning; Zhang, Heng; Zhang, Jie; Li, Zhong-Zheng; Duan, Wen-shan
2017-04-01
Damping dust acoustic waves described by the Korteweg-de Vries-type (KdV-type) equation and the nonlinear Schrödinger equation-type (quasi-NLSE) have been studied by the particle-in-cell (PIC) simulation method. The KdV-type equation and the quasi-NLSE with dust-neutral collision are analytically obtained by the reductive perturbation method. The PIC simulation methods for dust acoustic waves with damping force are shown. The PIC simulation results are compared with the analytical one. The relationship of the damping coefficient with the collision frequency is obtained. It is found that amplitudes of KdV-type solitary waves and quasienvelope solitary waves with damping force decrease exponentially.
Traffic jams, granular flow, and soliton selection
Kurtze, D.A.; Hong, D.C.
1995-07-01
The flow of traffic on a long section of road without entrances or exits can be modeled by continuum equations similar to those describing fluid flow. In a certain range of traffic density, steady flow becomes unstable against the growth of a cluster, or ``phantom`` traffic jam, which moves at a slower speed than the otherwise homogeneous flow. We show that near the onset of this instability, traffic flow is described by a perturbed Korteweg--de Vries (KdV) equation. The traffic jam can be identified with a soliton solution of the KdV equation. The perturbation terms select a unique member of the continuous family of KdV solitons. These results may also apply to the dynamics of granular relaxation.
Dressed ion-acoustic solitons in magnetized dusty plasmas
NASA Astrophysics Data System (ADS)
El-Labany, S. K.; El-Shamy, E. F.; El-Warraki, S. A.
2009-01-01
In the present research paper, the characteristics of ion acoustic solitary waves are investigated in hot magnetized dusty plasmas consisting of negatively charged dust grains, positively charged ion fluid, and isothermal electrons. Applying a reductive perturbation theory, a nonlinear Korteweg-de Vries (KdV) equation for the first-order perturbed potential and a linear inhomogeneous KdV-type equation for the second-order perturbed potentials are derived. Stationary solutions of these coupled equations are obtained using a renormalization method. The effects of the external oblique magnetic field, hot ion fluid, and higher-order nonlinearity on the nature of the ion acoustic solitary waves are discussed. The results complement and provide new insights into previously published results on this problem [R. S. Tiwari and M. K. Mishra, Phys. Plasmas 13, 062112 (2006)].
NASA Astrophysics Data System (ADS)
Tagare, S. G.
1997-09-01
It is found that a dusty plasma with inertial dust fluid and two-temperature isothermal ions admits both compressive and rarefactive solitary waves, as well as compressive and rarefactive double layers (depending on the concentration of low-temperature ions). In this paper, Korteweg-de Vries equation (KdV-type equations) with cubic and fourth-order nonlinearity at the critical density of low-temperature isothermal ions are derived to discuss properties of dust-acoustic solitary waves. In the vicinity of critical density of low-temperature ions, KdV-type equation with mixed nonlinearity is discussed. By using quasipotential analysis, critical Mach numbers M1c and M2c are obtained such that rarefactive dust-acoustic solitons exist when 1
The car following model considering traffic jerk
NASA Astrophysics Data System (ADS)
Ge, Hong-Xia; Zheng, Peng-jun; Wang, Wei; Cheng, Rong-Jun
2015-09-01
Based on optimal velocity car following model, a new model considering traffic jerk is proposed to describe the jamming transition in traffic flow on a highway. Traffic jerk means the sudden braking and acceleration of vehicles, which has a significant impact on traffic movement. The nature of the model is researched by using linear and nonlinear analysis method. A thermodynamic theory is formulated to describe the phase transition and critical phenomenon in traffic flow. The time-dependent Ginzburg-Landau (TDGL) equation and the modified Korteweg-de Vries (mKdV) equation are derived to describe the traffic flow near the critical point and the traffic jam. In addition, the connection between the TDGL and the mKdV equations are also given.
Ge, H X; Dai, S Q; Dong, L Y; Xue, Y
2004-12-01
An extended car following model is proposed by incorporating an intelligent transportation system in traffic. The stability condition of this model is obtained by using the linear stability theory. The results show that anticipating the behavior of more vehicles ahead leads to the stabilization of traffic systems. The modified Korteweg-de Vries equation (the mKdV equation, for short) near the critical point is derived by applying the reductive perturbation method. The traffic jam could be thus described by the kink-antikink soliton solution for the mKdV equation. From the simulation of space-time evolution of the vehicle headway, it is shown that the traffic jam is suppressed efficiently with taking into account the information about the motion of more vehicles in front, and the analytical result is consonant with the simulation one.
A model system for strong interaction between internal solitary waves
NASA Astrophysics Data System (ADS)
Bona, Jerry L.; Ponce, Gustavo; Saut, Jean-Claude; Tom, Michael M.
1992-01-01
A mathematical theory is mounted for a complex system of equations derived by Gear and Grimshaw that models the strong interaction of two-dimensional, long, internal gravity waves propagating on neighboring pycnoclines in a stratified fluid. For the model in question, the Cauchy problem is of interest, and is shown to be globally well-posed in suitably strong function spaces. Our results make use of Kato's theory for abstract evolution equations together with somewhat delicate estimates obtained using techniques from harmonic analysis. In weak function classes, a local existence theory is developed. The system is shown to be susceptible to the dispersive blow-up phenomenon investigated recently by Bona and Saut for Korteweg-de Vries-type equations.
From weak discontinuities to nondissipative shock waves
Garifullin, R. N. Suleimanov, B. I.
2010-01-15
An analysis is presented of the effect of weak dispersion on transitions from weak to strong discontinuities in inviscid fluid dynamics. In the neighborhoods of transition points, this effect is described by simultaneous solutions to the Korteweg-de Vries equation u{sub t}'+ uu{sub x}' + u{sub xxx}' = 0 and fifth-order nonautonomous ordinary differential equations. As x{sup 2} + t{sup 2} {yields}{infinity}, the asymptotic behavior of these simultaneous solutions in the zone of undamped oscillations is given by quasi-simple wave solutions to Whitham equations of the form r{sub i}(t, x) = tl{sub i} x/t{sup 2}.
ERIC Educational Resources Information Center
Bardell, Nicholas S.
2014-01-01
This paper describes how a simple application of de Moivre's theorem may be used to not only find the roots of a quadratic equation with real or generally complex coefficients but also to pinpoint their location in the Argand plane. This approach is much simpler than the comprehensive analysis presented by Bardell (2012, 2014), but it does not…
Demiray, Hilmi; Bayındır, Cihan
2015-09-15
In the present work, we consider the propagation of nonlinear electron-acoustic non-planar waves in a plasma composed of a cold electron fluid, hot electrons obeying a trapped/vortex-like distribution, and stationary ions. The basic nonlinear equations of the above described plasma are re-examined in the cylindrical coordinates through the use reductive perturbation method in the long-wave approximation. The modified cylindrical Korteweg-de Vries equation with fractional power nonlinearity is obtained as the evolution equation. Due to the nature of nonlinearity, which is fractional, this evolution equation cannot be reduced to the conventional Korteweg–de Vries equation. An analytical solution to the evolution equation, by use of the method developed by Demiray [Appl. Math. Comput. 132, 643 (2002); Comput. Math. Appl. 60, 1747 (2010)] and a numerical solution by employing a spectral scheme are presented and the results are depicted in a figure. The numerical results reveal that both solutions are in good agreement.
Nonlinear dynamics of soliton gas with application to "freak waves"
NASA Astrophysics Data System (ADS)
Shurgalina, Ekaterina
2017-04-01
So-called "integrable soliton turbulence" attracts much attention of scientific community nowadays. We study features of soliton interactions in the following integrable systems: Korteweg - de Vries equation (KdV), modified Korteweg - de Vries equation (mKdV) and Gardner equations. The polarity of interacted solitons dramatically influences on the process of soliton interaction. Thus if solitons have the same polarity the maximum of the wave field decreases during the process of nonlinear interactions as well statistical moments (skewness and kurtosis). In this case there is no abnormally large wave formation and this scenario is possible for all considered equation. Completely different results can be obtained for a soliton gas consisted of solitons with different polarities: such interactions lead to an increase of resulting impulse and kurtosis. Tails of distribution functions can grow significantly. Abnormally large waves (freak waves) appear in such solitonic fields. Such situations are possible just in case of mKdV and Gardner equations which admit the existence of bipolar solitons. New effect of changing a defect's moving direction in soliton lattices and soliton gas is found in the present study. Manifestation of this effect is possible as the result of negative phase shift of small soliton in the moment of nonlinear interaction with large solitons. It is shown that the effect of negative velocity is the same for KdV and mKdV equations and it can be found from the kinematic assumption without applying the kinetic theory. Averaged dynamics of the "smallest" soliton (defect) in a soliton gas, consisting of solitons with random amplitudes is investigated. The averaged criterion of velocity sign change confirmed by numerical simulation is obtained.
First Examples of de Vries-like Smectic A to Smectic C Phase Transitions in Ionic Liquid Crystals.
Kapernaum, Nadia; Müller, Carsten; Moors, Svenja; Schlick, M Christian; Wuckert, Eugen; Laschat, Sabine; Giesselmann, Frank
2016-12-15
In ionic liquid crystals, the orthogonal smectic A phase is the most common phase whereas the tilted smectic C phase is rather rare. We present a new study with five novel ionic liquid crystals exhibiting both a smectic A as well as the rare smectic C phase. Two of them have a phenylpyrimidine core whereas the other three are imidazolium azobenzenes. Their phase sequences and tilt angles were studied by polarizing microscopy and their temperature-dependent layer spacing as well as their translational and orientational order parameters were studied by X-ray diffraction. The X-ray tilt angles derived from X-ray studies of the layer contraction and the optically measured tilt angles of the five ionic liquid crystals were compared to obtain their de Vries character. Four of our five mesogens turned out to show de Vries-like behavior with a layer shrinkage that is far less than that expected for conventional materials. These materials can thus be considered as the first de Vries-type materials among ionic liquid crystals.
NASA Astrophysics Data System (ADS)
Senthil Kumar, V.; Kavitha, L.; Boopathy, C.; Gopi, D.
2017-10-01
Nonlinear interaction of electromagnetic solitons leads to a plethora of interesting physical phenomena in the diverse area of science that include magneto-optics based data storage industry. We investigate the nonlinear magnetization dynamics of a one-dimensional anisotropic ferromagnetic nanowire. The famous Landau-Lifshitz-Gilbert equation (LLG) describes the magnetization dynamics of the ferromagnetic nanowire and the Maxwell's equations govern the propagation dynamics of electromagnetic wave passing through the axis of the nanowire. We perform a uniform expansion of magnetization and magnetic field along the direction of propagation of electromagnetic wave in the framework of reductive perturbation method. The excitation of magnetization of the nanowire is restricted to the normal plane at the lowest order of perturbation and goes out of plane for higher orders. The dynamics of the ferromagnetic nanowire is governed by the modified Korteweg-de Vries (mKdV) equation and the perturbed modified Korteweg-de Vries (pmKdV) equation for the lower and higher values of damping respectively. We invoke the Hirota bilinearization procedure to mKdV and pmKdV equation to construct the multi-soliton solutions, and explicitly analyze the nature of collision phenomena of the co-propagating EM solitons for the above mentioned lower and higher values of Gilbert-damping due to the precessional motion of the ferromagnetic spin. The EM solitons appearing in the higher damping regime exhibit elastic collision thus yielding the fascinating state restoration property, whereas those of lower damping regime exhibit inelastic collision yielding the solitons of suppressed intensity profiles. The propagation of EM soliton in the nanoscale magnetic wire has potential technological applications in optimizing the magnetic storage devices and magneto-electronics.
NASA Astrophysics Data System (ADS)
Kraniotis, G. V.
2016-11-01
Exact solutions of the Klein-Gordon-Fock (KGF) general relativistic equation that describe the dynamics of a massive, electrically charged scalar particle in the curved spacetime geometry of an electrically charged, rotating Kerr-Newman-(anti) de Sitter black hole are investigated. In the general case of a rotating, charged, cosmological black hole the solution of the KGF equation with the method of separation of variables results in Fuchsian differential equations for the radial and angular parts which for most of the parameter space contain more than three finite singularities and thereby generalise the Heun differential equations. For particular values of the physical parameters (i.e. mass of the scalar particle) these Fuchsian equations reduce to the case of the Heun equation and the closed form analytic solutions we derive are expressed in terms of Heun functions. For other values of the parameters some of the extra singular points are false singular points. We derive the conditions on the coefficients of the generalised Fuchsian equation such that a singular point is a false point. In such a case the exact solution of the Fuchsian equation can in principle be simplified and expressed in terms of Heun functions. This is the generalisation of the case of a Heun equation with a false singular point in which the exact solution of Heun’s differential equation is expressed in terms of Gauß hypergeometric function. We also derive the exact solutions of the radial and angular equations for a charged massive scalar particle in the Kerr-Newman spacetime. The analytic solutions are expressed in terms of confluent Heun functions. Moreover, we derived the constraints on the parameters of the theory such that the solution simplifies and expressed in terms of confluent Kummer hypergeometric functions. We also investigate the radial solutions in the KN case in the regions near the event horizon and far from the black hole. Finally, we construct several expansions of the
The KdV—Burgers equation in a modified speed gradient continuum model
NASA Astrophysics Data System (ADS)
Lai, Ling-Ling; Cheng, Rong-Jun; Li, Zhi-Peng; Ge, Hong-Xia
2013-06-01
Based on the full velocity difference model, Jiang et al. put forward the speed gradient model through the micro-macro linkage (Jiang R, Wu Q S and Zhu Z J 2001 Chin. Sci. Bull. 46 345 and Jiang R, Wu Q S and Zhu Z J 2002 Trans. Res. B 36 405). In this paper, the Taylor expansion is adopted to modify the model. The backward travel problem is overcome by our model, which exists in many higher-order continuum models. The neutral stability condition of the model is obtained through the linear stability analysis. Nonlinear analysis shows clearly that the density fluctuation in traffic flow leads to a variety of density waves. Moreover, the Korteweg-de Vries—Burgers (KdV—Burgers) equation is derived to describe the traffic flow near the neutral stability line and the corresponding solution for traffic density wave is derived. The numerical simulation is carried out to investigate the local cluster effects. The results are consistent with the realistic traffic flow and also further verify the results of nonlinear analysis.
Few-cycle solitons in supercontinuum generation dynamics
NASA Astrophysics Data System (ADS)
Leblond, Hervé; Grelu, Philippe; Mihalache, Dumitru; Triki, Houria
2016-11-01
We review several propagation models that do not rely on the slowly-varying-envelope approximation (SVEA), and can thus be considered as fundamental models addressing the formation and propagation of few-cycle pulsed field structures and solitary waves arising in the course of intense ultrashort optical pulse evolution in nonlinear media and beyond octave-bandwidth optical spectrum broadening. These generic models are: the modified-Korteweg-de Vries (mKdV), the sine-Gordon (sG), and the mixed mKdV-sG equations. To include wave polarization dynamics, the vector extensions of both mKdV and sG equations are introduced. Multi-octave-spanning supercontinuum generation and few-cycle soliton structures are highlighted from numerical simulations.
Soliton-like thermophoresis of graphene wrinkles.
Guo, Yufeng; Guo, Wanlin
2013-01-07
We studied the thermophoretic motion of wrinkles formed in substrate-supported graphene sheets by nonequilibrium molecular dynamics simulations. We found that a single wrinkle moves along applied temperature gradient with a constant acceleration that is linearly proportional to temperature deviation between the heating and cooling sides of the graphene sheet. Like a solitary wave, the atoms of the single wrinkle drift upwards and downwards, which prompts the wrinkle to move forwards. The driving force for such thermophoretic movement can be mainly attributed to a lower free energy of the wrinkle back root when it is transformed from the front root. We establish a motion equation to describe the soliton-like thermophoresis of a single graphene wrinkle based on the Korteweg-de Vries equation. Similar motions are also observed for wrinkles formed in a Cu-supported graphene sheet. These findings provide an energy conversion mechanism by using graphene wrinkle thermophoresis.
Ion acoustic solitons in a solar wind magnetoplasma with Kappa distributed electrons
NASA Astrophysics Data System (ADS)
Devanandhan, Selvaraj; Singh, Satyavir; Singh Lakhina, Gurbax; Sreeraj, T.
2016-07-01
In many space plasma environments, the velocity distribution of particles often deviates from Maxwellian and is well-modelled by a kappa distribution function. We have analyzed the ion acoustic soliton in a magnetized consisting of plasma Protons, Helium ions, an electron beam and superthermal hot electrons following kappa distribution function. Under the assumption of weak nonlinearity, the ion-acoustic solitons are described by the Korteweg-de-Vries-Zakharov-Kuznetsov (KdV-ZK) equation. The solution of KdV-ZK equation is used to model the characteristics of the ion acoustic solitary waves in a solar wind magnetoplasma observed at 1 AU. We have found both slow and fast ion acoustic solitons in our study. It is found that the superthermality of hot electrons greatly influence the existence regime of the solitary waves. The numerical results of this study to explain solar wind observations will be discussed in detail.
Soliton and kink jams in traffic flow with open boundaries.
Muramatsu, M; Nagatani, T
1999-07-01
Soliton density wave is investigated numerically and analytically in the optimal velocity model (a car-following model) of a one-dimensional traffic flow with open boundaries. Soliton density wave is distinguished from the kink density wave. It is shown that the soliton density wave appears only at the threshold of occurrence of traffic jams. The Korteweg-de Vries (KdV) equation is derived from the optimal velocity model by the use of the nonlinear analysis. It is found that the traffic soliton appears only near the neutral stability line. The soliton solution is analytically obtained from the perturbed KdV equation. It is shown that the soliton solution obtained from the nonlinear analysis is consistent with that of the numerical simulation.
Nonlinear features of ion acoustic shock waves in dissipative magnetized dusty plasma
NASA Astrophysics Data System (ADS)
Sahu, Biswajit; Sinha, Anjana; Roychoudhury, Rajkumar
2014-10-01
The nonlinear propagation of small as well as arbitrary amplitude shocks is investigated in a magnetized dusty plasma consisting of inertia-less Boltzmann distributed electrons, inertial viscous cold ions, and stationary dust grains without dust-charge fluctuations. The effects of dissipation due to viscosity of ions and external magnetic field, on the properties of ion acoustic shock structure, are investigated. It is found that for small amplitude waves, the Korteweg-de Vries-Burgers (KdVB) equation, derived using Reductive Perturbation Method, gives a qualitative behaviour of the transition from oscillatory wave to shock structure. The exact numerical solution for arbitrary amplitude wave differs somehow in the details from the results obtained from KdVB equation. However, the qualitative nature of the two solutions is similar in the sense that a gradual transition from KdV oscillation to shock structure is observed with the increase of the dissipative parameter.
Nonlinear ion-acoustic waves in a degenerate plasma with nuclei of heavy elements
Hossen, M. A. Mamun, A. A.
2015-10-15
The ion-acoustic (IA) solitary waves propagating in a fully relativistic degenerate dense plasma (containing relativistic degenerate electron and ion fluids, and immobile nuclei of heavy elements) have been theoretically investigated. The relativistic hydrodynamic model is used to derive the Korteweg-de Vries (K-dV) equation by the reductive perturbation method. The stationary solitary wave solution of this K-dV equation is obtained to characterize the basic features of the IA solitary structures that are found to exist in such a degenerate plasma. It is found that the effects of electron dynamics, relativistic degeneracy of the plasma fluids, stationary nuclei of heavy elements, etc., significantly modify the basic properties of the IA solitary structures. The implications of this results in astrophysical compact objects like white dwarfs are briefly discussed.
Dressed electrostatic solitary waves in quantum dusty pair plasmas
NASA Astrophysics Data System (ADS)
Akbari-Moghanjoughi, M.
2010-05-01
Quantum-hydrodynamics model is applied to investigate the nonlinear propagation of electrostatic solitary excitations in a quantum dusty pair plasma. A Korteweg de Vries evolution equation is obtained using reductive perturbation technique and the higher-nonlinearity effects are derived by solving the linear inhomogeneous differential equation analytically using Kodama-Taniuti renormalizing method. The possibility of propagation of bright- and dark-type solitary excitations is examined. It is shown that a critical value of quantum diffraction parameter H exists, on either side of which, only one type of solitary propagation is possible. It is also found that unlike for the first-order amplitude component, the variation of H parameter dominantly affects the soliton amplitude in higher-order approximation. The effect of fractional quantum number density on compressive and rarefactive soliton dynamics is also discussed.
Nonlinear waves in coherently coupled Bose-Einstein condensates
NASA Astrophysics Data System (ADS)
Congy, T.; Kamchatnov, A. M.; Pavloff, N.
2016-04-01
We consider a quasi-one-dimensional two-component Bose-Einstein condensate subject to a coherent coupling between its components, such as realized in spin-orbit coupled condensates. We study how nonlinearity modifies the dynamics of the elementary excitations. The spectrum has two branches, which are affected in different ways. The upper branch experiences a modulational instability, which is stabilized by a long-wave-short-wave resonance with the lower branch. The lower branch is stable. In the limit of weak nonlinearity and small dispersion it is described by a Korteweg-de Vries equation or by the Gardner equation, depending on the value of the parameters of the system.
Nonlinear internal waves in shallow stratified lakes
NASA Astrophysics Data System (ADS)
Kurkina, Oxana; Talipova, Tatiana; Kurkin, Andrey; Ruvinskaya, Ekaterina; Pelinovsky, Efim
2015-04-01
Weakly nonlinear model of internal waves based on the extended Korteweg-de Vries equation - Gardner equation is applied to analyze possible shapes in shallow stratified lake - Sankhar Lake, Russia. Series of temperature variation in space and time are collected and analyzed. The spectra of such variations can be fitted by power function of frequency with exponent minus one, minus two. It is shown that temperature variations influence on kinematic characteristics of internal waves, mainly on the coefficient of quadratic nonlinearity. The solitary wave (soliton) of the first mode is an elevation wave with amplitude less 3 m (total depth of 15 m). The solitons of the second mode can have any polarity. Also the breathers of second mode can be generated in such lake.
Dressed electrostatic solitary waves in quantum dusty pair plasmas
Akbari-Moghanjoughi, M.
2010-05-15
Quantum-hydrodynamics model is applied to investigate the nonlinear propagation of electrostatic solitary excitations in a quantum dusty pair plasma. A Korteweg de Vries evolution equation is obtained using reductive perturbation technique and the higher-nonlinearity effects are derived by solving the linear inhomogeneous differential equation analytically using Kodama-Taniuti renormalizing method. The possibility of propagation of bright- and dark-type solitary excitations is examined. It is shown that a critical value of quantum diffraction parameter H exists, on either side of which, only one type of solitary propagation is possible. It is also found that unlike for the first-order amplitude component, the variation of H parameter dominantly affects the soliton amplitude in higher-order approximation. The effect of fractional quantum number density on compressive and rarefactive soliton dynamics is also discussed.