# General stationary solutions of the nonlocal nonlinear Schr\"{o}dinger equation and their relevance to the PT-symmetric system

## TL;DR

This paper derives general stationary solutions for the nonlocal nonlinear Schrödinger equation, including complex PT-symmetric solutions, and explores their implications for PT-symmetric physical systems.

## Contribution

It provides the first comprehensive set of stationary solutions for the NNLS equation, including complex PT-symmetric potentials and their properties.

## Key findings

- Derived general stationary solutions including elliptic and hyperbolic functions.
- Identified PT- and anti-PT-symmetric complex solutions with no spatial localization.
- Showed that PT-symmetric potentials do not undergo symmetry breaking in the linear case.

## Abstract

With the stationary solution assumption, we establish the connection between the nonlocal nonlinear Schr\"{o}dinger (NNLS) equation and an elliptic equation. Then, we obtain the general stationary solutions and discuss the relevance of their smoothness and boundedness to some integral constants. Those solutions, which cover the known results in the literature, include the unbounded Jacobi elliptic-function and hyperbolic-function solutions, the bounded sn-, cn- and dn-function solutions, as well as the hyperbolic soliton solutions. By the imaginary translation transformation of the NNLS equation, we also derive the complex-amplitude stationary solutions, in which all the bounded cases obey either the \PT- or anti-\PT-symmetric relation. In particular, the complex tanh-function solution can exhibit no spatial localization in addition to the dark and anti-dark soliton profiles, which is sharp contrast with the common dark soliton. Considering the physical relevance to \PT-symmetric system, we show that the complex-amplitude stationary solutions can yield a wide class of complex and time-independent \PT-symmetric potentials, and the symmetry breaking does not occur in the \PT-symmetric linear system with the associated potentials.