New stationary solutions of the cubic nonlinear Schr\"odinger equations for Bose-Einstein condensates

TL;DR

This paper extends a criterion for oscillation intervals to cubic nonlinear Schr"odinger equations, classifies solutions, and discovers new stationary solutions relevant to Bose-Einstein condensates, even with small nonlinearities.

Contribution

It analytically extends an oscillation criterion to nonlinear Schr"odinger equations and uncovers previously unnoticed stationary solutions in free-particle cases.

Findings

Discovered new stationary solutions in free-particle cases.

Solutions persist with external potentials.

Solutions appear even with small nonlinearities.

Abstract

We have previously formulated a simple criterion for deducing the intervals of oscillations in the solutions of second-order linear homogeneous differential equations. In this work, we extend analytically the same criterion to the cubic nonlinear Schr\"odinger equations that describe Bose-Einstein condensates. With this criterion guiding the search for solutions, we classify all types of solutions and we find new stationary solutions in the free-particle cases that were not noticed previously because of limited coverage in the adopted boundary conditions. The new solutions are produced by the nonlinear terms of the differential equations and they continue to exist when various external potentials are also incorporated. Surprisingly, these solutions appear when the nonlinearities are small.