Stationary solutions of the one-dimensional nonlinear Schroedinger equation: I. Case of repulsive nonlinearity

TL;DR

This paper analytically derives all stationary solutions of the one-dimensional nonlinear Schrödinger equation with repulsive nonlinearity, describing dark or grey soliton trains relevant to Bose-Einstein condensates and optical fibers.

Contribution

It provides a complete analytic characterization of stationary solutions for the repulsive case, including real and complex states, under various boundary conditions.

Findings

Real stationary states correspond to linear Schrödinger solutions.

Complex states are nonlinear, nodeless, and exhibit symmetry-breaking.

Solutions are applicable to Bose-Einstein condensates and optical fibers.

Abstract

All stationary solutions to the one-dimensional nonlinear Schroedinger equation under box and periodic boundary conditions are presented in analytic form. We consider the case of repulsive nonlinearity; in a companion paper we treat the attractive case. Our solutions take the form of stationary trains of dark or grey density-notch solitons. Real stationary states are in one-to-one correspondence with those of the linear Schr\"odinger equation. Complex stationary states are uniquely nonlinear, nodeless, and symmetry-breaking. Our solutions apply to many physical contexts, including the Bose-Einstein condensate and optical pulses in fibers.