Exact trapped $N$-soliton solutions of the nonlinear Schr\"odinger equation using the inverse problem method

TL;DR

This paper applies the inverse problem method to construct exact N-soliton solutions of the nonlinear Schrödinger equation in multiple dimensions, analyzing their stability and providing explicit forms for trapped solitons in BEC contexts.

Contribution

It introduces a systematic inverse problem approach to find exact trapped N-soliton solutions of the NLSE in various dimensions, including stability analysis for both attractive and repulsive interactions.

Findings

Stable solutions for g>0 (repulsive interactions)

Critical mass for instability in g<0 (attractive interactions)

Explicit forms of solutions using Gaussian and harmonic oscillator eigenfunctions

Abstract

In this work, we show the application of the ``inverse problem'' method to construct exact $N$ trapped soliton-like solutions of the nonlinear Schr\"odinger or Gross-Pitaevskii equation (NLSE and GPE, respectively) in one, two, and three spatial dimensions. This method is capable of finding the external (confining) potentials which render specific assumed waveforms exact solutions of the NLSE for both attractive ($g<0$) and repulsive ($g>0$) self-interactions. For both signs of $g$, we discuss the stability with respect to self-similar deformations and translations. For $g<0$, a critical mass $M_c$, or equivalently the number of particles, for instabilities to arise can often be found analytically. On the other hand, for the case with $g>0$ corresponding to repulsive self interactions which is often discussed in the atomic physics realm of Bose-Einstein condensates (BEC), the bound solutions are found to be always stable. For $g<0$, we also determine the critical mass numerically by using linear stability or Bogoliubov-de Gennes analysis, and compare these results with our analytic estimates. Various analytic forms for the trapped $N$-soliton solutions are discussed, including sums of Gaussians or higher-order eigenfunctions of the harmonic oscillator Hamiltonian.