Solutions of Nonlocal Equations Reduced from the AKNS Hierarchy

Abstract

In this paper, nonlocal reductions of the Ablowitz–Kaup–Newell–Suger (AKNS) hierarchy are collected, including the nonlocal nonlinear Schrödinger hierarchy, nonlocal modified Korteweg-de Vries hierarchy, and nonlocal versions of the sine-Gordon equation in nonpotential form. A reduction technique for solutions is employed, by which exact solutions in double Wronskian form are obtained for these reduced equations from those double Wronskian solutions of the AKNS hierarchy. As examples of dynamics, we illustrate new interaction of two-soliton solutions of the reverse-t nonlinear Schrödinger equation. Although as a single soliton, it is stationary that two solitons travel along completely symmetric trajectories in {x,t} plane and their amplitudes are affected by phase parameters. Asymptotic analysis is given as demonstration. The approach and relation described in this paper are systematic and general and can be used to other nonlocal equations.