Exact solvability of the Gross-Pitaevskii equation for bound states subjected to general potentials

TL;DR

This paper provides an exact analytical solution to the 1D Gross-Pitaevskii equation for bound states under various potentials, revealing new insights into its solvability and properties.

Contribution

It introduces a method to solve the nonlinear GP equation analytically for specific potentials by mapping it to a dynamical system and employing known functions.

Findings

Exact solutions for bound states in finite square wells and Dirac delta potentials.

Identification of conditions for stable solitons with repulsive interactions.

Connection between nonlinear GP solutions and linear layered potential problems.

Abstract

In this paper we present the analytic solution to the problem of bound states of the Gross-Pitaevskii (GP) equation in 1D and its properties, in the presence of external potentials in the form of finite square wells or attractive Dirac deltas, as well as stable solitons for repulsive defects. We show that the GP equation can be mapped to a first-order non-autonomous dynamical system, whose solutions can sometimes be written in terms of known functions. The formal solutions of this non-conservative system can be written with the help of Glauber-Trotter formulas or a series of ordered exponentials in the coordinate $x$. With this we illustrate how to solve any nonlinear problem based on a construction due to Mello and Kumar for the linear case (layered potentials). For the benefit of the reader, we comment on the difference between the integrability of a quantum system and the solvability of the wave equation.