Solutions of local and nonlocal discrete complex modified Korteweg-de Vries equations and continuum limits

TL;DR

This paper derives local and nonlocal discrete complex modified Korteweg-de Vries equations from the discrete Ablowitz-Kaup-Newell-Segur framework, constructs their solutions, and explores their continuum limits.

Contribution

It introduces new local and nonlocal discrete complex mKdV equations via the Cauchy matrix approach and analyzes their solutions and continuum limits.

Findings

Explicit soliton and Jordan-block solutions are obtained.

The dynamical behavior of 1-soliton solutions is analyzed.

Continuum limits connect discrete equations to their continuous counterparts.

Abstract

Cauchy matrix approach for the discrete Ablowitz-Kaup-Newell-Segur equations is reconsidered, where two `proper' discrete Ablowitz-Kaup-Newell-Segur equations and two `unproper' discrete Ablowitz-Kaup-Newell-Segur equations are derived. The `proper' equations admit local reduction, while the `unproper' equations admit nonlocal reduction. By imposing the local and nonlocal complex reductions on the obtained discrete Ablowitz-Kaup-Newell-Segur equations, two local and nonlocal discrete complex modified Korteweg-de Vries equations are constructed. For the obtained local and nonlocal discrete complex modified Korteweg-de Vries equations, soliton solutions and Jordan-block solutions are presented by solving the determining equation set. The dynamical behaviors of 1-soliton solution are analyzed and illustrated. Continuum limits of the resulting local and nonlocal discrete complex modified Korteweg-de Vries equations are discussed.