  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.
