Bilinearization-reduction approach to the classical and nonlocal semi-discrete modified Korteweg-de Vries equations with nonzero backgrounds

TL;DR

This paper develops explicit quasi double Casoratian solutions for semi-discrete modified Korteweg-de Vries equations with nonzero backgrounds, using bilinearization and reduction techniques, enabling detailed analysis of soliton and periodic dynamics.

Contribution

It introduces a bilinearization-reduction method to derive explicit solutions for classical and nonlocal semi-discrete mKdV equations with nonzero backgrounds, advancing solution construction techniques.

Findings

Explicit quasi double Casoratian solutions derived.

Solutions encompass solitonic, periodic, and rational types.

Dynamics analyzed for focusing and defocusing cases.

Abstract

Quasi double Casoratian solutions are derived for a bilinear system reformulated from the coupled semi-discrete modified Korteweg-de Vries equations with nonzero backgrounds. These solutions, when applied with the classical and nonlocal reduction techniques, also satisfy the corresponding classical and nonlocal semi-discrete modified Korteweg-de Vries equations with nonzero backgrounds. They can be expressed explicitly, allowing for an easy investigation of the dynamics of systems. As illustrative examples, the dynamics of solitonic, periodic and rational solutions with a plane wave background are examined for the focusing semi-discrete Korteweg-de Vries equation and the defocusing reverse-space-time complex semi-discrete Korteweg-de Vries equation.