Integrable nonlocal nonlinear equations

TL;DR

This paper introduces new classes of integrable nonlocal nonlinear equations, including reverse space-time and reverse time variants, expanding the understanding of PT-symmetric and nonlocal integrable systems with explicit solutions and conservation laws.

Contribution

It presents novel reverse space-time and reverse time nonlocal integrable equations derived from symmetry reductions, along with their Lax pairs, conservation laws, and explicit soliton solutions.

Findings

Introduced reverse space-time and reverse time nonlocal integrable equations.

Constructed Lax pairs and found explicit one-soliton solutions.

Established conservation laws and inverse scattering transforms for these systems.

Abstract

A nonlocal nonlinear Schr\"odinger (NLS) equation was recently found by the authors and shown to be an integrable infinite dimensional Hamiltonian equation. Unlike the classical (local) case, here the nonlinearly induced "potential" is $PT$ symmetric thus the nonlocal NLS equation is also $PT$ symmetric. In this paper, new {\it reverse space-time} and {\it reverse time} nonlocal nonlinear integrable equations are introduced. They arise from remarkably simple symmetry reductions of general AKNS scattering problems where the nonlocality appears in both space and time or time alone. They are integrable infinite dimensional Hamiltonian dynamical systems. These include the reverse space-time, and in some cases reverse time, nonlocal nonlinear Schr\"odinger, modified Korteweg-deVries (mKdV), sine-Gordon, $(1+1)$ and $(2+1)$ dimensional three-wave interaction, derivative NLS, "loop soliton", Davey-Stewartson (DS), partially $PT$ symmetric DS and partially reverse space-time DS equations. Linear Lax pairs, an infinite number of conservation laws, inverse scattering transforms are discussed and one soliton solutions are found. Integrable reverse space-time and reverse time nonlocal discrete nonlinear Schr\"odinger type equations are also introduced along with few conserved quantities. Finally, nonlocal Painlev\'e type equations are derived from the reverse space-time and reverse time nonlocal NLS equations.