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

TL;DR

The paper develops a quadrature-based framework to solve the 1D Gross-Pitaevskii equation for bound states in general external potentials, including Dirac-delta defects and finite wells. By modeling the potential as a continuous limit of layered Dirac deltas, it maps the GP problem to a first-order nonautonomous dynamical system and expresses solutions as ordered exponentials, yielding a spectral equation $\mathcal{F}(E)=0$ for bound-state energies. Explicit nonlinear bound states are constructed for delta and square-well potentials using Jacobi elliptic functions, and the approach is extended to arbitrary potentials via a layered construction and Glauber-Trotter-like factorization. The work clarifies the distinction between solvability and integrability in 1D quantum systems, and provides a formal, exact-solvability framework for nonlinear wave equations with general potentials, with the practical task focused on computing the spectral function $\mathcal{F}(E)$.

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.

Figures (12)

• Figure 1: Roots of the polynomials that define the elliptic Jacobi functions.
• Figure 2: Bright solitons in repulsive barrier. $\phi_{\alpha>0}(0)$ and $\alpha$ are fixed. When $\gamma \rightarrow 0$, $\phi_{\bar{\alpha}>0}(-\bar{\tau})=\phi_{\bar{\alpha}>0}(\bar{\tau}) \rightarrow \infty$ and $\bar{\tau} \rightarrow \infty$.
• Figure 3: Bright solitons. Repulsive defect and negative coupling. The bound state is repelled but not destroyed by the point-like potential.
• Figure 4: Bright solitons. Attractive defect and negative coupling. Both contributions make the binding mechanism stronger.
• Figure 5: Wave functions computed with a logarithmic quadrature, valid for positive nonlinear coupling. When the densities are properly normalized, the localization length of $\gamma=0$ and $\gamma>0$ can be compared, and the loosely bound state can be identified by its larger width.