Multiplicity, asymptotics and stability of standing waves for nonlinear Schrödinger equation with rotation

Abstract

In this article, we study the multiplicity, asymptotics and stability of standing waves with prescribed mass $c>0$ for nonlinear Schrödinger equation with rotation in the mass-supercritical regime arising in Bose-Einstein condensation. Under suitable restriction on the rotation frequency, by searching critical points of the corresponding energy functional on the mass-sphere, we obtain a local minimizer $u_c$ and a mountain pass solution ${\hat{u}}_c$. %under suitable assumptions on the related parameters. Furthermore, we show that $u_c$ is a ground state for small mass $c>0$ and describe a mass collapse behavior of the minimizers as $c\to 0$, while ${\hat{u}}_c$ is an excited state. Finally, we prove that the standing wave associated with $u_c$ is stable. Notice that the pioneering works \cite{aMsC,shYZ} imply that finite time blow-up of solutions to this model occurred in the mass-supercritical setting, therefore, we in the present paper obtain a new stability result. The main contribution of this paper is to extend the main results in \cite{JeSp,gYlW} concerning the same model from mass-subcritical and mass-critical regimes to mass-supercritical regime, where the physically most relevant case is covered.

arXiv comment20 pages,comments are welcome!

References in corpus (13)

• Vortex formation in a stirred Bose-Einstein condensate
• Vortices in a Bose-Einstein Condensate
• Rapidly Rotating Atomic Gases
• Bosons in anisotropic traps: ground state and vortices
• Stationary states of a rotating Bose-Einstein condensate: routes to vortex nucleation
• On dipolar quantum gases in the unstable regime
• Vortices in attractive Bose-Einstein condensates in two dimensions
• Ground State Asymptotics of a Dilute, Rotating Gas
• Scattering for nonlinear Schrodinger equation under partial harmonic confinement
• The Nonexistence of Vortices for Rotating Bose-Einstein Condensates with Attractive Interactions
• Stability and instability properties of rotating Bose-Einstein condensates
• Split-merge cycle, fragmented collapse, and vortex disintegration in rotating Bose-Einstein condensates with attractive interactions
• Center of mass rotation and vortices in an attractive Bose gas