Integrable nonlocal complex mKdV equation: soliton solution and gauge equivalence

TL;DR

This paper establishes the gauge equivalence between the nonlocal complex mKdV equation and a spin-like model, and constructs Darboux transformations to derive various exact solutions including solitons and periodic waves.

Contribution

It demonstrates the gauge equivalence of the nonlocal complex mKdV equation to a spin model and develops Darboux transformations for explicit solution construction.

Findings

Nonlocal complex mKdV is gauge equivalent to a spin-like model

Constructed Darboux transformations for the nonlocal equation

Derived explicit solutions: dark, W-type, M-type solitons, and periodic solutions

Abstract

In this paper, we prove that the nonlocal complex modified Korteweg-de Vries (mKdV) equation introduced by Ablowitz and Musslimani [Nonlinearity, 29, 915-946 (2016)] is gauge equivalent to a spin-like model. From the gauge equivalence, one can see that there exists significant difference between the nonlocal complex mKdV equation and the classical complex mKdV equation. Through constructing the Darboux transformation(DT) for nonlocal complex mKdV equation, a variety of exact solutions including dark soliton, W-type soliton, M-type soliton and periodic solutions are derived.