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arXiv:2502.06120

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

Mirón, M., Emerson Sadurní

Published Feb 10, 2025• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2502.06120

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In this work, we discuss an application of the "inverse problem" method to find the external trapping potential, which has particular N trapped soliton-like solutions of the Gross-Pitaevskii equation (GPE) also known as the cubic nonlinear Schrödinger equation (NLSE). This inverse method assumes particular forms for the trapped soliton wave function, which then determines the (unique) external (confining) potential. The latter renders these assumed waveforms exact solutions of the GPE (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 Mc 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, 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 in one, two, and three spatial dimensions are discussed, including sums of Gaussians or higher-order eigenfunctions of the harmonic oscillator Hamiltonian.
The Gross-Pitaevskii equation is solved by analytic methods for an external double-well potential that is an infinite square well plus a δδδ-function central barrier. We find solutions that have the symmetry of the non-interacting Hamiltonian as well as asymmetric solutions that bifurcate from the symmetric solutions for attractive interactions and from the antisymmetric solutions for repulsive interactions. We present a variational approximation to the asymmetric state as well as an approximate numerical approach. Stability of the states is briefly considered.
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Abstract The Gross-Pitaevskii equation is solved by analytic methods for anexternal double-well potential that is an infinite square well plusa δ-function central barrier. We find solutions that havethe symmetry of the non-interacting Hamiltonian as well as asymmetricsolutions that bifurcate from the symmetric solutions for attractive interactions and from the antisymmetric solutions for repulsive interactions. We presenta variational approximation to the asymmetric state as well as anapproximate numerical approach. Stability of the states is briefly considered.
We study the stationary nonlinear Schr\"odinger equation, or Gross-Pitaevskii equation, for a one-dimensional finite square-well potential. By neglecting the mean-field interaction outside the potential well it is possible to discuss the transport properties of the system analytically in terms of ingoing and outgoing waves. Resonances and bound states are obtained analytically. The transmitted flux shows a bistable behavior. Novel crossing scenarios of eigenstates similar to beak-to-beak structures are observed for a repulsive mean-field interaction. It is proven that resonances transform to bound states due to an attractive nonlinearity and vice versa for a repulsive nonlinearity, and the critical nonlinearity for the transformation is calculated analytically. The bound-state wave functions of the system satisfy an oscillation theorem as in the case of linear quantum mechanics. Furthermore, the implications of the eigenstates on the dymamics of the system are discussed.