Solutions of local and nonlocal equations reduced from the AKNS hierarchy

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 Schrödinger 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 Schrödinger 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.

26 pages, 4 figures

References in corpus (2)

• Integrable $\mathcal{P}T$-symmetric local and nonlocal vector nonlinear Schrödinger equations: a unified two-parameter model
• Bilinearisation-reduction approach to the nonlocal discrete nonlinear Schrödinger equations

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• $(2+1)$-dimensional AKNS($-N$) Systems: $N=3,4$
• Hirota bilinear forms of the AKNS($N$) systems
• Local and nonlocal complex discrete and semi-discrete sine-Gordon equations and solutions