Lax Integrability and Soliton Solutions for a Nonisospectral Integrodifferential System

Searching for integrable systems and constructing their exact solutions are of both theoretical and practical value. In this paper, Ablowitz–Kaup–Newell–Segur (AKNS) spectral problem and its time evolution equation are first generalized by embedding a new spectral parameter. Based on the generalized AKNS spectral problem and its time evolution equation, Lax integrability of a nonisospectral integrodifferential system is then verified. Furthermore, exact solutions of the nonisospectral integrodifferential system are formulated through the inverse scattering transform (IST) method. Finally, in the case of reflectionless potentials, the obtained exact solutions are reduced ton-soliton solutions. Whenn=1andn=2, the characteristics of soliton dynamics of one-soliton solutions and two-soliton solutions are analyzed with the help of figures.

Download Full-text

The Eigenfunctions and Exact Solutions of Discrete mKdV Hierarchy with Self-Consistent Sources via the Inverse Scattering Transform

Advances in Applied Mathematics and Mechanics ◽

10.4208/aamm.2013.m450 ◽

2015 ◽

Vol 7 (5) ◽

pp. 663-674 ◽

Cited By ~ 1

Author(s):

Q. Li ◽

J. B. Zhang ◽

D. Y. Chen

Keyword(s):

Exact Solutions ◽

Spectral Problem ◽

Inverse Scattering ◽

Soliton Solutions ◽

Inverse Scattering Transform ◽

The Self ◽

Scattering Transform ◽

Self Consistent ◽

Exact Soliton Solutions ◽

Mkdv Hierarchy

AbstractAnother form of the discrete mKdV hierarchy with self-consistent sources is given in the paper. The self-consistent sources is presented only by the eigenfunctions corresponding to the reduction of the Ablowitz-Ladik spectral problem. The exact soliton solutions are also derived by the inverse scattering transform.

Download Full-text

Derivation and soliton dynamics of a new non-isospectral and variable-coefficient system

Thermal Science ◽

10.2298/tsci180510076x ◽

2019 ◽

Vol 23 (Suppl. 3) ◽

pp. 639-646

Author(s):

Bo Xu ◽

Sheng Zhang

Keyword(s):

Variable Coefficient ◽

Soliton Solutions ◽

Inverse Scattering Transform ◽

Inelastic Collisions ◽

Time Varying ◽

Transform Method ◽

Soliton Dynamics ◽

Scattering Transform ◽

Isospectral Problem ◽

Time Evolution Equation

Under investigation in this paper is a new and more general non-isospectral and variable-coefficient non-linear integrodifferential system. Such a system is Lax integrable because of its derivation from the compatibility condition of a generalized linear non-isospectral problem and its accompanied time evolution equation which is generalized in this paper by embedding four arbitrary smooth enough functions. Soliton solutions of the derived system are obtained in the framework of the inverse scattering transform method with a time-varying spectral parameter. It is graphically shown the dynamical evolutions of the obtained soliton solutions possess time-varying amplitudes and that the inelastic collisions can happen between two-soliton solutions.

Download Full-text

Lax Integrability and Exact Solutions of a Variable-Coefficient and Nonisospectral AKNS Hierarchy

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2016-0191 ◽

2018 ◽

Vol 19 (3-4) ◽

pp. 251-262 ◽

Cited By ~ 3

Author(s):

Sheng Zhang ◽

Siyu Hong

Keyword(s):

Exact Solutions ◽

Variable Coefficient ◽

Soliton Solutions ◽

Inverse Scattering Transform ◽

Time Dependent ◽

Time Varying ◽

Spatial Structures ◽

Transform Method ◽

Scattering Transform ◽

Time Evolution Equation

AbstractIn this paper, a variable-coefficient and nonisospectral Ablowitz–Kaup–Newell–Segur (vcniAKNS) hierarchy with Lax integrability is constructed by embedding a finite number of differentiable and time-dependent functions into the well-known AKNS spectral problem and its time evolution equation. In the framework of inverse scattering transform method with time-varying spectral parameter, the constructed vcniAKNS hierarchy is solved exactly. As a result, exact solutions and their reduced n-soliton solutions of the vcniAKNS hierarchy are obtained. It is graphically shown that the parity of an embedded time-dependent function has connection with the symmetrical characteristics of the spatial structures and singular points of the obtained one-soliton solutions.

Download Full-text

Inverse scattering transform for a new non-isospectral integrable non-linear AKNS model

Thermal Science ◽

10.2298/tsci17s1153g ◽

2017 ◽

Vol 21 (suppl. 1) ◽

pp. 153-160 ◽

Cited By ~ 2

Author(s):

Xudong Gao ◽

Sheng Zhang

Keyword(s):

Evolution Equation ◽

Exact Solutions ◽

Inverse Scattering ◽

Spectral Parameter ◽

Soliton Solutions ◽

Transformation Method ◽

Linear Partial Differential Equations ◽

Scattering Transform ◽

Non Linear ◽

Isospectral Problem

Constructing integrable systems and solving non-linear partial differential equations are important and interesting in non-linear science. In this paper, Ablowitz-Kaup-Newell-Segur (AKNS)?s linear isospectral problem and its accompanied time evolution equation are first generalized by embedding a new non-isospectral parameter whose varying with time obeys an arbitrary smooth enough function of the spectral parameter. Based on the generalized AKNS linear problem and its evolution equation, a new non-isospectral Lax integrable non-linear AKNS model is then derived. Furthermore, exact solutions of the derived AKNS model is obtained by extending the inverse scattering transformation method with new time-varying spectral parameter. In the case of reflectinless potentials, explicit n-soliton solutions are finally formulated through the obtained exact solutions.

Download Full-text

Solutions to τ-equation hierarchy related to the AKNS spectral problem: Addendum to “The exact solutions for the nonisospectral AKNS hierarchy through the inverse scattering transform”

Physica A Statistical Mechanics and its Applications ◽

10.1016/j.physa.2005.03.038 ◽

2005 ◽

Vol 357 (3-4) ◽

pp. 415-426 ◽

Cited By ~ 1

Author(s):

Tong-Ke Ning ◽

Deng-Yuan Chen ◽

Da-Jun Zhang

Keyword(s):

Exact Solutions ◽

Spectral Problem ◽

Inverse Scattering ◽

Inverse Scattering Transform ◽

Akns Hierarchy ◽

Scattering Transform

Download Full-text

Inverse scattering transform for a supersymmetric Korteweg-de Vries equation

Thermal Science ◽

10.2298/tsci180512081z ◽

2019 ◽

Vol 23 (Suppl. 3) ◽

pp. 677-684

Author(s):

Sheng Zhang ◽

Caihong You

Keyword(s):

Exact Solutions ◽

Inverse Scattering ◽

Evolution Equations ◽

Vries Equation ◽

Soliton Solutions ◽

Grassmann Algebra ◽

Inverse Scattering Transform ◽

Linear Evolution ◽

Scattering Transform ◽

De Vries

In this paper, the inverse scattering transform is extended to a super Korteweg-de Vries equation with an arbitrary variable coefficient by using Kulish and Zeitlin?s approach. As a result, exact solutions of the super Korteweg-de Vries equation are obtained. In the case of reflectionless potentials, the obtained exact solutions are reduced to soliton solutions. More importantly, based on the obtained results, an approach to extending the scattering transform is proposed for the supersymmetric Korteweg-de Vries equation in the 1-D Grassmann algebra. It is shown the the approach can be applied to some other supersymmetric non-linear evolution equations in fluids.

Download Full-text

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

Mathematical Problems in Engineering ◽

10.1155/2019/4058041 ◽

2019 ◽

Vol 2019 ◽

pp. 1-10 ◽

Cited By ~ 2

Author(s):

Yuxin Lin ◽

Yong Fang ◽

Huanhe Dong

Keyword(s):

Spectral Problem ◽

Inverse Scattering ◽

Soliton Solutions ◽

Type Equation ◽

Lax Pair ◽

Inverse Scattering Transform ◽

Nls Equation ◽

Nonlinear Schrödinger ◽

Scattering Transform ◽

Matrix Spectral Problem

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.

Download Full-text