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Advances in Nonlinear Analysis

Standing waves for Choquard equation with noncritical rotation

Yi-cen Mao, Jie Yang, Yu Su

Published Jan 1, 2024• 1 citation
Publisher

Abstract

Abstract We investigate the existence and stability of standing waves with prescribed mass c > 0 c\gt 0 for Choquard equation with noncritical rotation in Bose-Einstein condensation. Then, we consider the mass collapse behavior of standing waves, the ratio of energy to mass and the Lagrange multiplier, as c → 0 + c\to {0}^{+} . Our results extend the existing results.

DOI
10.1515/anona-2023-0140

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Abstract We investigate the existence and stability of standing waves with prescribed mass c > 0 c\gt 0 for Choquard equation with noncritical rotation in Bose-Einstein condensation. Then, we consider the mass collapse behavior of standing waves, the ratio of energy to mass and the Lagrange multiplier, as c → 0 + c\to {0}^{+} . Our results extend the existing results.
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Abstract. We study the existence and stability / instability properties of standing waves for a nonlinear Schrödinger equation arising in dipolar Bose-Einstein condensate in the unstable regime. Two cases are studied: the first when the system is free, the second when gradually a trapping potential is add. In both cases this leads to the search of critical points of a constrained functional which is unbounded from below on the constraint. In the first case we prove the existence and instability of ground states by showing that the constrained functional has a so-called mountain pass geometry. In the second case we prove that, if the trapping potential is small, two different kind of standing waves appears: one corresponds to a local minima of the constrained energy functional, the other is again of mountain pass type. Despite the problem is mass supercritical and the functional unbounded from below the standing waves associated to the local minima are orbitally stable. Eventually we show that the addition of the trapping potential, however small, create a “gap ” in the ground state energy level of the system. 1.
We present a new family of stationary solutions to the cubic nonlinear Schrödinger equation with an elliptic function potential. In the limit of a sinusoidal potential our solutions model a quasi-one-dimensional dilute gas Bose-Einstein condensate trapped in a standing light wave. Provided that the ratio of the height of the variations of the condensate to its dc offset is small enough, both trivial phase and nontrivial phase solutions are shown to be stable. Recent developments allow for experimental investigation of these predictions.
We study the nonlinear Schrödinger equation arising in dipolar Bose--Einstein condensate in the unstable regime. Two cases are studied: the first when the system is free, the second when gradually a trapping potential is added. In both cases we first focus on the existence and stability/instability properties of standing waves. Our approach leads to the search of critical points of a constrained functional which is unbounded from below on the constraint. In the free case, by showing that the constrained functional has a so-called mountain pass geometry, we prove the existence of standing states with least energy, the ground states, and show that any ground state is orbitally unstable. Moreover, when the system is free, we show that small data in the energy space scatter in all regimes, stable and unstable. In the second case, if the trapping potential is small, we prove that two different kinds of standing waves appear: one corresponds to a topological local minimizer of the constrained energy functional and consists in ground states, and the other is again of mountain pass type but now corresponds to excited states. We also prove that any ground state is a topological local minimizer. Despite the problem being mass supercritical and the functional being unbounded from below, the standing waves associated to the set of ground states turn out to be orbitally stable. Actually, from the physical point of view, the introduction of the trapping potential stabilizes the system that is initially unstable. Related to this we observe that it also creates a gap in the ground state energy level of the system. In addition when the trapping potential is active the presence of standing waves with arbitrary small norm does not permit small data scattering. Eventually some asymptotic results are also given.