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

Abstract

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.

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

“…This is due to the novel spectral parameter embedded into the linear spectral problem associated to the AKNS equations. Though there are similar studies [29][30][31][32][33], the AKNS equations, the spectral parameter and the results presented in this paper are different from those in literature. Like the Taylor series method [35,36], the IST method, which is also called as the non-linear Fourier analysis, plays an important role in non-linear science and two scale thermodynamics [37,38].…”

“…and the similar manipulations [31,33], we can determine the time-dependence of the scattering data of the spectral problem (3): x r x respectively.…”

“…Thanks to the powerful VIM, HPM, and EXP method proposed by Ji-Huan He, a large number of exact and approximate solutions have been obtained in different fields. Recently, Zhang and Hong [21] extended the IST method to the generalized isospectral Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy, the variable-coefficient non-isospectral --------------Toda lattice hierarchy [22], the generalized non-isospectral AKNS hierarchies [23][24][25][26][27], the mixed spectral KdV hierarchy with self-consistent sources [28] and the mixed spectral AKNS hierarchies [29][30][31][32][33], supersymmetric KdV equation [34], respectively.…”

An Integrable Evolution System and Its Analytical Solutions with the Help of Mixed Spectral AKNS Matrix Problem

“…Figures 2, 3 and 6-9 are obtained. Compared with the mixed spectral AKNS hierarchy [7] mentioned earlier in Equation (28) and the other results in [15][16][17][18], the work of this paper has some differences. Specifically, Equation (11) or its operator form (25) are different from the following equations [15][16][17][18]:…”

Exact Solutions of Nonlinear Equations in Mathematical Physics via Negative Power Expansion Method

“…Constructing exact solutions of nonlinear mathematical physical equations is of theoretical and practical significance. Since the famous Korteweg-de Vries (KdV) equation was solved in 1967 [7], a large number of exact solutions like [2][3][4][5][6][7]10,11,[13][14][15][16][18][19][20][22][23][24][25][26][27][28]31,33,34,36,38,41,43] of nonlinear partial differential equations (PDEs) have been found. The exp-function method [10] proposed by He and Wu has been widely used for constructing exact solutions of nonlinear PDEs.…”