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

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

Inverse scattering transform for new mixed spectral Ablowitz-Kaup-Newell-Segur equations

Thermal Science ◽

10.2298/tsci2004437z ◽

2020 ◽

Vol 24 (4) ◽

pp. 2437-2444

Author(s):

Sheng Zhang ◽

Caihong You

Keyword(s):

Inverse Scattering ◽

Spectral Parameter ◽

Analytical Methods ◽

Linear Problem ◽

Soliton Solutions ◽

Inverse Scattering Transform ◽

Time Varying ◽

Spatial Structures ◽

New Exact Solutions ◽

Scattering Transform

The inverse scattering transform plays a very important role in promoting the development of analytical methods to solve non-linear PDE exactly. In this paper, new and more general mixed spectral Ablowitz-Kaup-Newell-Segur equations are derived and solved by embedding a novel time-varying spectral parameter in-to an associated linear problem and the inverse scattering transform. As a result, new exact solutions and n-soliton solutions are obtained. To gain more insights into the embedded spectral parameter and the obtained solutions, some dynamical evolutions, and spatial structures are simulated. It is shown that the derived Ablowitz-Kaup-Newell-Segur equations are Lax integrable and the obtained soliton solutions possess time-varying amplitudes.

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

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

Mixed spectral AKNS hierarchy from linear isospectral problem and its exact solutions

Open Physics ◽

10.1515/phys-2015-0040 ◽

2015 ◽

Vol 13 (1) ◽

Cited By ~ 4

Author(s):

Sheng Zhang ◽

Xu-Dong Gao

Keyword(s):

Exact Solutions ◽

Inverse Scattering ◽

Variable Coefficient ◽

Soliton Solutions ◽

Variable Coefficients ◽

Necessary Condition ◽

Scattering Problems ◽

Systematic Analysis ◽

Akns Hierarchy ◽

Isospectral Problem

AbstractIn this paper, the AKNS isospectral problem and its corresponding time evolution are generalized by embedding three coefficient functions. Starting from the generalizedAKNS isospectral problem, a mixed spectralAKNS hierarchy with variable coefficients is derived. Thanks to the selectivity of these coefficient functions, the mixed spectral AKNS hierarchy contains not only isospectral equations but also nonisospectral equations. Based on a systematic analysis of the related direct and inverse scattering problems, exact solutions of the mixed spectral AKNS hierarchy are obtained through the inverse scattering transformation. In the case of reflectionless potentials, the obtained exact solutions are reduced to n-soliton solutions. This paper shows that the AKNS spectral problem being nonisospectral is not a necessary condition to construct a nonisospectral AKNS hierarchy and that the inverse scattering transformation can be used for solving some other variable-coefficient mixed hierarchies of isospectral equations and nonisospectral equations.

Download Full-text

Lax Integrability and Soliton Solutions for a Nonisospectral Integrodifferential System

Complexity ◽

10.1155/2017/9457078 ◽

2017 ◽

Vol 2017 ◽

pp. 1-10 ◽

Cited By ~ 4

Author(s):

Sheng Zhang ◽

Siyu Hong

Keyword(s):

Evolution Equation ◽

Exact Solutions ◽

Spectral Problem ◽

Time Evolution ◽

Soliton Solutions ◽

Inverse Scattering Transform ◽

Integrodifferential System ◽

Soliton Dynamics ◽

Scattering Transform ◽

Time Evolution Equation

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