Prolongation structures of a generalized coupled Korteweg-de Vries equation and Miura transformation

Cao, Yuanhao; Wang, Deng‐Shan

doi:10.1016/j.cnsns.2009.10.020

Cited by 16 publications

(3 citation statements)

References 11 publications

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“…This method is one of the effective skills to verify whether the equation is integrable or not. The theory has been widely applied to many integrable equations, such as coupled KdV equation [32,33], coupled nonlinear Schröinger equation [34], and Heisenberg ferromagnet equation [35].…”

Section: Introductionmentioning

confidence: 99%

Prolongation Structures and N‐Soliton Solutions for a New Nonlinear Schrödinger‐Type Equation via Riemann‐Hilbert Approach

Lin

Fang

Dong

2019

Mathematical Problems in Engineering

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In this paper, a new integrable nonlinear Schrödinger-type (NLST) equation is investigated by prolongation structures theory and Riemann-Hilbert (R-H) approach. Via prolongation structures theory, the Lax pair of the NLST equation, a 2×2 matrix spectral problem, is derived. Depending on the analysis of red the spectral problem, a R-H problem of the NLST equation is formulated. Furthermore, through a specific R-H problem with the vanishing scattering coefficient, N-soliton solutions of the NLST equation are expressed explicitly. Moreover, a few key differences are presented, which exist in the implementation of the inverse scattering transform for NLST equation and cubic nonlinear Schrödinger (NLS) equation. Finally, the dynamic behaviors of soliton solutions are shown by selecting appropriate spectral parameter λ, respectively.

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Section: Introductionmentioning

confidence: 99%

Prolongation Structures and N‐Soliton Solutions for a New Nonlinear Schrödinger‐Type Equation via Riemann‐Hilbert Approach

Lin

Fang

Dong

2019

Mathematical Problems in Engineering

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“…In 1895, Korteweg and de Vries [1][2] derived a nonlinear evolution equation as follows: expansion, Bäcklund transformation, algebraic method and so on [1][2][3][4][5][6][7]. Using the inverse scattering method, we could obtain the N-soliton solution of KdV equation (see Fig.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Evolution Equations for Second-order Spectral Problem

Liu¹,

Yuan²,

Wang³

2012

JCP

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Soliton equations are infinite-dimensional integrable systems described by nonlinear evolution equations. As one of the soliton equations, long wave equation takes on profound significance of theory and reality. By using the method of nonlinearization, the relation between long wave equation and second-order eigenvalue problem is generated. Based on the nonlinearized Lax pairs, Euler-Lagrange function and Legendre transformations, a reasonable Jacobi-Ostrogradsky coordinate system is obtained. Moreover, by means of the Bargmann constrained condition between the potential function and the eigenfunction, the Lax pairs is equivalent to matrix spectral problem. Furthermore, the involutive representations of the solutions for long wave equation are generated.

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“…The nonlinearization approach (or constrained flows) of eigenvalue problems or Lax Pairs has been used to seek the relations between the infinite and finite integrable system. There are series of finite integrable system obtained by this approach [8][9][10][11][12][13][14][15][16][17][18][19]. Lagrange function and the Legendre transformations have been derived, and a reasonable Jacobi-Ostrogradsky coordinate system has been found.…”

Section: Introductionmentioning

confidence: 99%

Integrability of the Reduction Fourth-Order Eigenvalue Problem

Wang¹,

Liu²,

Yuan³

2012

JCP

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To study the reduced fourth-order eigenvalue problem, the Bargmann constraint of this problem has been given, and the associated Lax pairs have been nonlineared. By means of the viewpoint of Hamilton mechanics, the Euler-Lagrange function and the Legendre transformations have been derived, and a reasonable Jacobi-Ostrogradsky coordinate system has been found. Then, the Hamiltonian cannonical coordinate system equivalent to this eigenvalue problem has been obtained on the symplectic manifolds. It is proved to be an infinite-dimensional integrable Hamilton system in the Liouville sense. Moreover the involutive representation of the solutions is generated for the evolution equations hierarchy in correspondence with this reduced fourth-order eigenvalue problem.

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Prolongation structures of a generalized coupled Korteweg-de Vries equation and Miura transformation

Cited by 16 publications

References 11 publications

Prolongation Structures and N‐Soliton Solutions for a New Nonlinear Schrödinger‐Type Equation via Riemann‐Hilbert Approach

Prolongation Structures and N‐Soliton Solutions for a New Nonlinear Schrödinger‐Type Equation via Riemann‐Hilbert Approach

Nonlinear Evolution Equations for Second-order Spectral Problem

Integrability of the Reduction Fourth-Order Eigenvalue Problem

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