Cauchy matrix solutions to some local and nonlocal complex equations

TL;DR

This paper introduces a Cauchy matrix reduction method to derive solutions for various local and nonlocal complex equations, including soliton and Jordan block solutions, with analysis of their dynamics.

Contribution

It develops a novel Cauchy matrix reduction technique to obtain explicit solutions for local and nonlocal complex equations from original systems.

Findings

Derived Cauchy matrix-type soliton solutions.

Presented Jordan block solutions for complex equations.

Analyzed the dynamical behaviors of solutions graphically.

Abstract

In this paper, we develop a Cauchy matrix reduction technique that enables us to obtain solutions for the reduced local and nonlocal complex equations from the Cauchy matrix solutions of the original before-reduction systems. Specifically, by imposing local and nonlocal complex reductions on some Ablowitz-Kaup-Newell-Segur-type equations, we study some local and nonlocal complex equations, involving the local and nonlocal complex modified Korteweg-de Vries equation, the local and nonlocal complex sine-Gordon equation, the local and nonlocal potential nonlinear Schr\"{o}dinger equation and the local and nonlocal potential complex modified Korteweg-de Vries equation. Cauchy matrix-type soliton solutions and Jordan block solutions for the aforesaid local and nonlocal complex equations are presented. The dynamical behaviors of some obtained solutions are analyzed with graphical illustrations.