Solitons and breathers for nonisospectral mKdV equation with Darboux transformation

TL;DR

This paper develops Darboux transformation techniques for a nonisospectral, variable coefficient mKdV equation, enabling the construction of diverse soliton and breather solutions and exploring their properties for potential soliton management.

Contribution

It introduces a simplified integrability condition for the nonisospectral vc-mKdV equation and constructs its Darboux transformation, leading to new solution types and insights into variable coefficient effects.

Findings

Generated double-breather and periodical soliton-breather solutions

Analyzed effects of variable coefficients on soliton properties

Provided conditions for soliton amplitude, polarity, velocity, and width control

Abstract

Under investigation in this paper is the nonisospectral and variable coefficients modified Kortweg-de Vries (vc-mKdV) equation, which manifests in diverse areas of physics such as fluid dynamics, ion acoustic solitons and plasma mechanics. With the degrees of restriction reduced, a simplified constraint is introduced, under which the vc-mKdV equation is an integrable system and the spectral flow is time-varying. The Darboux transformation for such equation is constructed, which gives rise to the generation of variable kinds of solutions including the double-breather coherent structure, periodical soliton-breather and localized solitons and breathers. In addition, the effect of variable coefficients and initial phases is discussed in terms of the soliton amplitude, polarity, velocity and width, which might provide feasible soliton management with certain conditions taken into account.