Integration of the matrix KdV equation with self-consistent source

Bondarenko, Natalia P.; Freiling, Gerhard; Urazboev, G. U.

doi:10.1016/j.chaos.2013.02.010

Cited by 7 publications

(9 citation statements)

References 23 publications

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“…Here ω t can be absorbed by a redefinition of either q or r, which exchanges P and Q in the corresponding linear equation due to a necessary application of (4.3). The scalar version appeared in [3], also see [33][34][35][36]. Here we chose β(t) = sin(3t), p 1 = 1, p 2 = −1/2, a 1 = b 1 = (1, 0) and a 2 = (0, 1) = b 2 in (4.6).…”

Section: Matrix Kdv Equation With Self-consistent Sourcesmentioning

confidence: 99%

Self-Consistent Sources for Integrable Equations Via Deformations of Binary Darboux Transformations

2016

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We reveal the origin and structure of self-consistent source extensions of integrable equations from the perspective of binary Darboux transformations. They arise via a deformation of the potential that is central in this method. As examples, we obtain in particular matrix versions of self-consistent source extensions of the KdV, Boussinesq, sine-Gordon, nonlinear Schrödinger, KP, Davey-Stewartson, two-dimensional Toda lattice and discrete KP equation. We also recover a (2+1)-dimensional version of the Yajima-Oikawa system from a deformation of the pKP hierarchy. By construction, these systems are accompanied by a hetero binary Darboux transformation, which generates solutions of such a system from a solution of the source-free system and additionally solutions of an associated linear system and its adjoint. The essence of all this is encoded in universal equations in the framework of bidifferential calculus.

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Section: Matrix Kdv Equation With Self-consistent Sourcesmentioning

confidence: 99%

Self-Consistent Sources for Integrable Equations Via Deformations of Binary Darboux Transformations

2016

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“…One of the problems for future interest consists in looking for the corresponding analogues of these solutions in the obtained generalizations. The same question concerns lumps and rogue wave solutions that were investigated in several integrable systems recently [3,20,21,22,23,24,62,81] It is also known that inverse scattering and spectral methods [1,7,41,57] were applied to generate solutions of equations with self-consistent sources [6,27,46]. An extension of these methods to the obtained hierarchies and comparison with results that can be provided by BDTs (e.g., following [67]) presents an interest for us.…”

Section: Resultsmentioning

confidence: 99%

“…KPSCS and the respective matrix (1 + 1)-dimensional counterpart (KdV equation with selfconsistent sources) have been investigated recently via Darboux transformations [45,83] and the inverse scattering method [6]. b) c 1 = 0, c 2 = 1.…”

Section: Introductionmentioning

confidence: 99%

Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy

Chvartatskyi¹

2015

SIGMA

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Abstract. New extensions of the KP and modified KP hierarchies with self-consistent sources are proposed. The latter provide new generalizations of (2+1)-dimensional integrable equations, including the DS-III equation and the N -wave problem. Furthermore, we recover a system that contains two types of the KP equation with self-consistent sources as special cases. Darboux and binary Darboux transformations are applied to generate solutions of the proposed hierarchies.

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“…It is well known that the soliton equations with self-consistent sources (SESCSs) can exhibit abundant nonlinear dynamics compared to soliton equations themselves and have important physical applications [1]. These SESCSs are usually used to describe interactions between different solitary waves and are relevant in some problems related with hydrodynamics, solid state physics, or plasma physics [2][3][4]. The sources appear in solitary waves with nonconstant velocity and lead to a variety of dynamics of physical models [2].…”

Section: Introductionmentioning

confidence: 99%

“…These SESCSs are usually used to describe interactions between different solitary waves and are relevant in some problems related with hydrodynamics, solid state physics, or plasma physics [2][3][4]. The sources appear in solitary waves with nonconstant velocity and lead to a variety of dynamics of physical models [2]. For example, the KdV equation with a self-consistent source (KdV-SCS) describes the interaction of long and short capillary-gravity waves [5,6].…”

Section: Introductionmentioning

confidence: 99%

Solutions and Painlevé Property for the KdV Equation with Self-Consistent Source

Shen

Yao

2018

Mathematical Problems in Engineering

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In this paper, the polynomial solutions in terms of Jacobi’s elliptic functions of the KdV equation with a self-consistent source (KdV-SCS) are presented. The extended (G′/G)-expansion method is utilized to obtain exact traveling wave solutions of the KdV-SCS, which finally are expressed in terms of the hyperbolic function, the trigonometric function, and the rational function. Meanwhile we find the Lie point symmetry and Lie symmetry group and give several group-invariant solutions for the KdV-SCS. Finally, we supplement the results of the Painlevé property in our previous work and get the Bäcklund transformations of the KdV-SCS.

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Integration of the matrix KdV equation with self-consistent source

Cited by 7 publications

References 23 publications

Self-Consistent Sources for Integrable Equations Via Deformations of Binary Darboux Transformations

Self-Consistent Sources for Integrable Equations Via Deformations of Binary Darboux Transformations

Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy

Solutions and Painlevé Property for the KdV Equation with Self-Consistent Source

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