Weakly nonlinear analysis of resonant PDEs in recent literature has generated a number of resonant systems for slow evolution of the normal mode amplitudes that possess remarkable properties. Despite being infinite-dimensional Hamiltonian systems with cubic nonlinearities in the equations of motion, these resonant systems admit special analytic solutions, which furthermore display periodic perfect energy returns to the initial configurations. Here, we construct a very large class of resonant systems that shares these properties that have so far been seen in specific examples emerging from a few standard equations of mathematical physics (the Gross-Pitaevskii equation, nonlinear wave equations in Anti-de Sitter spacetime). Our analysis provides an additional conserved quantity for all of these systems, which has been previously known for the resonant system of the two-dimensional Gross-Pitaevskii equation, but not for any other cases.
The Lowest Landau Level (LLL) equation emerges as an accurate approximation for a class of dynamical regimes of Bose-Einstein Condensates (BEC) in two-dimensional isotropic harmonic traps in the limit of weak interactions. Building on recent developments in the field of spatially confined extended Hamiltonian systems, we find a fully nonlinear solution of this equation representing periodically modulated precession of a single vortex. Motions of this type have been previously seen in numerical simulations and experiments at moderately weak coupling. Our work provides the first controlled analytic prediction for trajectories of a single vortex, suggests new targets for experiments, and opens up the prospect of finding analytic multi-vortex solutions.Since the discovery of Bose-Einstein condensates (BEC) in ultracold atomic gases, considerable experimental and theoretical work has been devoted to their properties in the presence of rotation, which leads to formation of quantized vortices (for reviews, see [1][2][3]). While certain stationary configurations of vortices have been a subject of semi-analytic and analytic investigations [4][5][6], to the best of our knowledge results on nonlinear motions of vortices have so far involved either numerics or approximations [7][8][9][10][11][12]. In this article, we show that analytic progress can be made by drawing inspiration from recent developments in the field of spatially confined extended Hamiltonian systems [13][14][15][16], which have not thus far surfaced in the BEC literature. As a first step, we find analytic solutions describing periodically modulated precession of a single vortex. We believe that a systematic generalization of our approach will eventually be used to study multi-vortex dynamics, a question of great appeal from both phenomenological and mathematical perspective.In situations relevant for us here, one considers Bose-Einstein condensates narrowly confined in one spatial direction, so that the dynamics is effectively two-dimensional. In this two-dimensional xy-plane, the condensate is placed in an isotropic harmonic potential known as the 'trap.' The system is described by the Gross-Pitaevskii (GP) equation for the condensate wavefunction Ψ(t, x, y)where g is a dimensionless coupling constant, proportional to the atomic scattering length and the total number of atoms (we impose |Ψ| 2 dxdy = 1). Our focus will be on studying this equation in the weakly nonlinear regime 0 < g 1 [17]. Positions of condensate vortices are given by the zeroes of Ψ.A key feature for the weakly nonlinear dynamics of (1) is that the eigenmodes of the linearized problem (g = 0) oscillate with integer frequencies, and consequently arbitrarily small nonlinearities produce significant effects over long times, due to the presence of resonances. To deal with this situation, the time-averaging method [18] is particularly suitable. One starts by going to the interaction picture, which amounts to expanding Ψ in the formwhere e imφ χ nm (r) are normalized isotropic harmonic oscilla...
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.
We numerically investigate spherically symmetric collapses in the Gross-Pitaevskii equation with attractive nonlinearity in a harmonic potential. Even below threshold for direct collapse, the wave function bounces off from the origin and may eventually become singular after a number of oscillations in the trapping potential. This is reminiscent of the evolution of Einstein gravity sourced by a scalar field in Anti-de Sitter space where collapse corresponds to black hole formation. We carefully examine the long time evolution of the wave function for continuous families of initial states in order to sharpen out this qualitative coincidence which may bring new insights in both directions. On one hand, we comment on possible implications for the so-called Bosenova collapses in cold atom Bose-Einstein condensates. On the other hand, Gross-Pitaevskii provides a toy model to study the relevance of either the resonance conditions or the nonlinearity for the problem of Anti-de Sitter instability.
We study periodically driven scalar fields and the resulting geometries with global AdS asymptotics. These solutions describe the strongly coupled dynamics of dual finite-size quantum systems under a periodic driving which we interpret as Floquet condensates. They span a continuous two-parameter space that extends the linearized solutions on AdS. We map the regions of stability in the solution space. In a significant portion of the unstable subspace, two very different endpoints are reached depending upon the sign of the perturbation. Collapse into a black hole occurs for one sign. For the opposite sign instead one attains a regular solution with periodic modulation. We also construct quenches where the driving frequency and amplitude are continuously varied. Quasistatic quenches can interpolate between pure AdS and sourced solutions with time periodic vev. By suitably choosing the quasistatic path one can obtain boson stars dual to Floquet condensates at zero driving field. We characterize the adiabaticity of the quenching processes. Besides, we speculate on the possible connections of this framework with time crystals.
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