Lax Integrability and Soliton Solutions for a Nonisospectral Integrodifferential System

Abstract

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 to -soliton solutions. When = 1 and = 2, the characteristics of soliton dynamics of one-soliton solutions and two-soliton solutions are analyzed with the help of figures.

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

“…Starting from eqs. (1) and (2) equipped with the parameter ik t = 0.5 + ik -2k 2 , Zhang and Hong [11] derived a non-isospectral integrodifferential system:…”

“…In soliton theory, the non-isospectral PDE are a kind of non-linear equations describing the solitary waves in a certain type of non-uniform media, while the isospectral PDE often describe solitary waves in lossless and uniform media. Recently, the investigation on derivations and solutions of non-isospectral PDE has attached much attentions [2][3][4][5][6][7][8][9][10][11][12][13][14][15]. In 2017, by introducing a new spectral parameter ik t = 0.5 + ik -2k 2 , Zhang and Hong [11] generalized the linear isospectral problem [3]:…”

Analytical insights into three models: Exact solutions and nonlinear vibrations

“…where h and n are determined by equations (11) to (13), and FðnÞ satisfies equation (9). In what follows, from equation 14we derive some special solutions of equation (1).…”

“…Considering a special case of equation 13, we set n ¼ qðxÞt þ pðxÞ (15) where pðxÞ and qðxÞ are undetermined functions of x. Then we can easily see from the first term h t of equations (11), (13) and (15) that h ttt ¼ 0. Thus, we further suppose that…”

“…In the past several decades, many effective methods for exactly solving nonlinear PDEs have been presented. [2][3][4][5][6][7][8][9][10][11][12][13] Among them, the exp-function method 7 proposed by He and Wu is an effective mathematical tool for constructing wave solutions, solitary solutions and periodic solutions. Recently, the analytical investigation of nonlinear vibrations has received considerable attention.…”