Cauchy matrix solutions to some local and nonlocal complex equations

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.