Inverse scattering transform for the nonlocal nonlinear Schr\"{o}dinger equation with nonzero boundary conditions

TL;DR

This paper develops an inverse scattering transform method for the nonlocal nonlinear Schrödinger equation with nonzero boundary conditions, analyzing scattering data, symmetries, and soliton solutions in various cases.

Contribution

It extends the inverse scattering transform to nonlocal NLS equations with nonzero boundary conditions, providing explicit soliton solutions and analyzing different boundary cases.

Findings

Explicit 1-soliton solutions are derived.

Two 2-soliton solutions are constructed.

In one case, no solitons exist.

Abstract

In 2013 a new nonlocal symmetry reduction of the well-known AKNS scattering problem was found; it was shown to give rise to a new nonlocal $PT$ symmetric and integrable Hamiltonian nonlinear Schr\"{o}dinger (NLS) equation. Subsequently, the inverse scattering transform was constructed for the case of rapidly decaying initial data and a family of spatially localized, time periodic one soliton solution were found. In this paper, the inverse scattering transform for the nonlocal NLS equation with nonzero boundary conditions at infinity is presented in the four cases when the data at infinity have constant amplitudes. The direct and inverse scattering problems are analyzed. Specifically, the direct problem is formulated, the analytic properties of the eigenfunctions and scattering data and their symmetries are obtained. The inverse scattering problem is developed via a left-right Riemann-Hilbert problem in terms of a suitable uniformization variable and the time dependence of the scattering data is obtained. This leads to a method to linearize/solve the Cauchy problem. Pure soliton solutions are discussed and explicit 1-soliton solution and two 2-soliton solutions are provided for three of the four different cases corresponding to two different signs of nonlinearity and two different values of the phase difference between plus and minus infinity. In the one other case there are no solitons.