2D Toda lattice equation with self-consistent sources: Casoratian type solutions, bilinear Bäcklund transformation and Lax pair

Wang, Hongyan; Hu, Xing−Biao; Gegenhasi,

doi:10.1016/j.cam.2005.08.052

Cited by 23 publications

(22 citation statements)

References 31 publications

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“…Differentiating with respect to y, this becomes the two-dimensional Toda lattice equation (ln(1 + V )) xy = V + − 2V + V − [52][53][54]. A self-consistent source extension, and corresponding exact solutions, has been obtained in [55] (also see [56]), using the framework of Hirota's bilinear difference operators [54]. (9.1) leads to a matrix version.…”

Section: Matrix Two-dimensional Toda Lattice Equation With Self-consimentioning

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 Two-dimensional Toda Lattice Equation With Self-consimentioning

confidence: 99%

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

2016

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“…Such coupled systems of integrable couplings can also provide concrete examples of soliton equations exhibiting solution diversity (see, e.g. [35][36][37][38][39][40][41][42][43]). There remains, however, an open question [44].…”

Section: Applicable Analysis 467mentioning

confidence: 99%

Component-trace identities for Hamiltonian structures

2010

Applicable Analysis

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We show that on a particular class of semi-direct sums of matrix Lie algebras, component traces of the matrix product can produce bilinear forms which are non-degenerate, symmetric and invariant under the Lie product. The corresponding variational identities are called componenttrace identities and provide tools in generating Hamiltonian structures of integrable couplings including the perturbation equations. An illustrative example of applying component-trace identities is given for the KdV hierarchy.

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“…The investigation of exact solutions of NLPDEs plays an important role in the study of these phenomena such as the nonlinear dynamics and the mechanism behind the phenomena. With the development of soliton theory, many powerful methods for obtaining exact solutions of NLPDEs have been presented, such as inverse scattering transformation [1], Hirota bilinear method [2], Bäcklund transformation [3], Darboux transformation [4], homotopy perturbation method [5], extended Riccati equation rational expansion method [6], asymptotic methods [7], extended auxiliary function method [8], algebraic method [9], Jacobi elliptic function expansion method [10],and so on [11][12][13].…”

Section: Introductionmentioning

confidence: 99%

New Exact Analytical Solutions for the General KdV Equation with Variable Coefficients

Hong

2014

MCA

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Abstract-In this paper, a general algebraic method based on the generalized Jacobi elliptic functions expansion method, the improved general mapping deformation method and the extended auxiliary function method with computerized symbolic computation is proposed to construct more new exact solutions of a generalized KdV equation with variable coefficients. As a result, eight families of new generalized Jacobi elliptic function wave solutions and Weierstrass elliptic function solutions of the equation are obtained by using this method, some of these solutions are degenerated to soliton-like solutions, trigonometric function solutions in the limit cases when the modulus of the Jacobi elliptic functions 1 m  or 0, which shows that the general method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.

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2D Toda lattice equation with self-consistent sources: Casoratian type solutions, bilinear Bäcklund transformation and Lax pair

Cited by 23 publications

References 31 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

Component-trace identities for Hamiltonian structures

New Exact Analytical Solutions for the General KdV Equation with Variable Coefficients

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