Solutions of local and nonlocal equations reduced from the AKNS hierarchy

TL;DR

This paper explores local and nonlocal reductions of the AKNS hierarchy, deriving exact solutions and analyzing soliton interactions, including novel symmetric two-soliton dynamics in nonlocal nonlinear Schrödinger equations.

Contribution

It introduces a systematic reduction technique to obtain explicit double Wronskian solutions for various local and nonlocal integrable equations derived from the AKNS hierarchy.

Findings

Exact double Wronskian solutions for reduced equations

New symmetric two-soliton interaction patterns

Asymptotic analysis of soliton dynamics

Abstract

In the paper possible local and nonlocal reductions of the Ablowitz-Kaup-Newell-Suger (AKNS) hierarchy are collected, including the Korteweg-de Vries (KdV) hierarchy, modified KdV hierarchy and their nonlocal versions, nonlinear Schr\"{o}dinger hierarchy and their nonlocal versions, sine-Gordon equation in nonpotential form and its nonlocal forms. 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 Schr\"{o}dinger equation. Although as a single soliton it is always stationary, 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.