General soliton solution to a nonlocal nonlinear Schr\"odinger equation with zero and nonzero boundary conditions

TL;DR

This paper develops a comprehensive method to derive general soliton solutions for a PT-symmetric nonlocal nonlinear Schrödinger equation under both zero and nonzero boundary conditions, expanding understanding of nonlocal integrable systems.

Contribution

It introduces a unified approach combining Hirota's method and KP hierarchy reduction to construct explicit soliton solutions for the nonlocal NLS equation with PT-symmetry.

Findings

Constructed general N-soliton solutions with zero boundary conditions.

Derived soliton solutions with nonzero boundary conditions from single KP hierarchy.

Provided explicit forms of all possible soliton solutions for the nonlocal NLS equation.

Abstract

General soliton solutions to a nonlocal nonlinear Schr\"odinger (NLS) equation with PT-symmetry for both zero and nonzero boundary conditions {are considered} via the combination of Hirota's bilinear method and the Kadomtsev-Petviashvili (KP) hierarchy reduction method. First, general $N$-soliton solutions with zero boundary conditions are constructed. Starting from the tau functions of the two-component KP hierarchy, it is shown that they can be expressed in terms of either Gramian or double Wronskian determinants. On the contrary, from the tau functions of single component KP hierarchy, general soliton solutions to the nonlocal NLS equation with nonzero boundary conditions are obtained. All possible soliton solutions to nonlocal NLS with Parity (PT)-symmetry for both zero and nonzero boundary conditions are found in the present paper.