Darboux Transformation for the Nonisospectral and Variable-coefficient KdV Equation

TL;DR

This paper develops a Darboux transformation method for the nonisospectral, variable-coefficient KdV equation, enabling the derivation of multi-soliton solutions and analyzing their dynamics in nonuniform media.

Contribution

It introduces a Darboux transformation tailored for the nonisospectral, variable-coefficient KdV equation, providing explicit soliton solutions and insights into their dynamics.

Findings

Multi-soliton solutions derived explicitly.

Spectral parameters influence soliton behavior.

Conditions for soliton management are established.

Abstract

With the nonuniform media taken into account, the nonisospectral and variable-coefficient Korteweg-de Vries equation, which describes various physical situations such as fluid dynamics and plasma, is under investigation in this paper. With appropriate selection of wave functions, the Darboux transformation is constructed, by which the multi-soliton solutions are derived and graphs are presented. The spectral parameters, coefficients and initial phase are discussed analytically and numerically to demonstrate their respective effect on the soliton dynamics, which plays a role in achieving the feasible soliton management with explicit conditions taken into account.