Cauchy matrix solutions of some local and nonlocal complex equations

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 nonreduced 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 Schrödinger 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 behavior of some of the obtained solutions is analyzed with graphical illustrations.