Inverse scattering transform for the nonlocal reverse space-time Sine-Gordon, Sinh-Gordon and nonlinear Schr\"{o}dinger equations with nonzero boundary conditions

TL;DR

This paper develops the inverse scattering transform for nonlocal reverse space-time Sine-Gordon, Sinh-Gordon, and nonlinear Schrödinger equations with nonzero boundary conditions, enabling explicit soliton solutions and analysis of their properties.

Contribution

It extends the inverse scattering transform framework to nonlocal equations with nonzero boundary conditions, addressing branching issues and providing explicit soliton solutions.

Findings

Constructed IST for nonlocal equations with NZBCs

Derived explicit 1- and 2-soliton solutions

Analyzed effects of boundary phase and nonlinearity signs

Abstract

The reverse space-time (RST) Sine-Gordon, Sinh-Gordon and nonlinear Schr\"odinger equations were recently introduced and shown to be integrable infinite-dimensional dynamical systems. The inverse scattering transform (IST) for rapidly decaying data was also constructed. In this paper, IST for these equations with nonzero boundary conditions (NZBCs) at infinity is presented. The NZBC problem is more complicated due to the associated branching structure of the associated linear eigenfunctions. With constant amplitude at infinity, four cases are analyzed; they correspond to two different signs of nonlinearity and two different values of the phase at infinity. Special soliton solutions are discussed and explicit 1-soliton and 2-soliton solutions are found. In terms of IST, the difference between the RST Sine-Gordon/Sinh-Gordon equations and the RST NLS equation is the time dependence of the scattering data. Spatially dependent boundary conditions are also briefly considered.