Stationary solutions of the one-dimensional nonlinear Schroedinger equation: II. Case of attractive nonlinearity

TL;DR

This paper analytically characterizes all stationary solutions of the one-dimensional nonlinear Schrödinger equation with attractive nonlinearity, revealing new nonlinear phenomena and symmetry-breaking states relevant to Bose-Einstein condensates.

Contribution

It provides a complete set of analytic solutions for attractive nonlinearity, including novel nodeless classes and symmetry-breaking states, expanding understanding of nonlinear quantum systems.

Findings

Solutions form bounded, quantized trains of bright solitons

Identification of two unique nonlinear nodeless solution classes

Prediction of experimental phenomena in Bose-Einstein condensates

Abstract

All stationary solutions to the one-dimensional nonlinear Schroedinger equation under box or periodic boundary conditions are presented in analytic form for the case of attractive nonlinearity. A companion paper has treated the repulsive case. Our solutions take the form of bounded, quantized, stationary trains of bright solitons. Among them are two uniquely nonlinear classes of nodeless solutions, whose properties and physical meaning are discussed in detail. The full set of symmetry-breaking stationary states are described by the $C_{n}$ character tables from the theory of point groups. We make experimental predictions for the Bose-Einstein condensate and show that, though these are the analog of some of the simplest problems in linear quantum mechanics, nonlinearity introduces new and surprising phenomena.