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

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.

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

2018 ◽  
Vol 19 (3-4) ◽  
pp. 251-262 ◽  
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.

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

2017 ◽  
Vol 21 (suppl. 1) ◽  
pp. 153-160 ◽  
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.

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

2015 ◽  
Vol 7 (5) ◽  
pp. 663-674 ◽  
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.

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

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.

Sign in / Sign up

Export Citation Format

Share Document