Two infinite families of resonant solutions for the Gross-Pitaevskii equation

Abstract

We consider the two-dimensional Gross-Pitaevskii equation describing a Bose-Einstein condensate in an isotropic harmonic trap. In the small coupling regime, this equation is accurately approximated over long times by the corresponding nonlinear resonant system whose structure is determined by the fully resonant spectrum of the linearized problem. We focus on two types of consistent truncations of this resonant system: first, to sets of modes of fixed angular momentum, and second, to excited Landau levels. Each of these truncations admits a set of explicit analytic solutions with initial conditions parametrized by three complex numbers. Viewed in position space, the fixed angular momentum solutions describe modulated oscillations of dark rings, while the excited Landau level solutions describe modulated precession of small arrays of vortices and antivortices. We place our findings in the context of similar results for other spatially confined nonlinear Hamiltonian systems in recent literature.

Energy-level splitting for weakly interacting bosons in a harmonic trap

“…We finally comment on the 'quantum Gross-Pitaevskii equations' [53], a hierarchy of equations inspired by variational tensor network techniques and interpolating between the classical Gross-Pitaevskii equation and quantum bosons with contact interactions described by (1). Since the operation of enhanced symmetries in the weak coupling regime has by now been seen both in the Gross-Pitaevskii equation [12,20,[37][38][39] and in the full quantum system in our present work, one expects it to have manifestations within the entire quantum Gross-Pitaevskii hierarchy. It would be interesting to investigate this subject further.…”

“…A number of resonant systems emerging from such weakly nonlinear considerations [20,22,23,25] possess features closely reminiscent of what we have used above to analyze the resonant LLL Hamiltonian. In [41], these features have been distilled into a formulation of a very large class of resonant systems that, in particular, admit an analog of the Zconservation that has played a crucial role in the derivations of this section.…”

“…With this sign factor, the wavefunctions can be conveniently transformed in a manner identical to the considerations of [20] using identities for Laguerre polynomials. Using the explicit expressions for the Laguerre polynomials,…”

“…This classical Hamiltonian has been studied in the literature [12,20] as the resonant approximation to the Gross-Pitaevskii equation, which is the classical limit of (1) -see also [11] where the same resonant Hamiltonian emerges from approximating a different related PDE. Similar resonant Hamiltonian systems emerge as approximations to other physically motivated PDEs, see [22,23,[25][26][27][28][29].…”

Fully resonant scalars on asymptotically AdS wormholes

“…Equispaced spectra play an important role in turbulent energy transfer leading to the non-perturbative AdS instability [14]- [19]. This leads to a rich phenomenology that also appears in non-linear models of different physical nature as in self-gravitating scalars on a spherical cavity in 3+1 [20], on systems describing Bose-Einstein condensates [21], and vortex precession [22], as well as in the conformal dynamics on the Einstein Universe [23]. A program to classify all the spacetimes in which a minimally coupled scalar probe may lead to an exactly solvable effective Schroedinger problem, suggestively dubbed "Klein-Gordonization", was initiated in [24].…”