Solutions of three nonlocal equations with self-consistent sources by the inverse scattering transform and reductions

Based on the Lax pairs and inverse scattering theory, we propose a reduction method by which we naturally reduce the AKNS hierarchy with self-consistent sources to several nonlocal nonlinear integrable hierarchies with self-consistent sources. The key is the properties of the squared eigenfunctions and scattering data associated with the AKNS scattering problems under symmetry conditions, and reducing the number of sources by half. By the reductions, we derive three nonlocal hierarchies including the nonlocal nonlinear Schrödinger hierarchy with self-consistent sources, the nonlocal complex modified Korteweg–de Vries hierarchy with self-consistent sources, and the nonlocal modified Korteweg–de Vries hierarchy with self-consistent sources, as well as their soliton solutions. As an example, we describe the shape and motion of a one-soliton solution of the nonlocal modified Korteweg–de Vries equation with self-consistent sources and compare it with its counterpart without sources. This reduction method can be applied to both nonlocal and classical (local) reductions of the AKNS hierarchy with self-consistent sources.