Existence and stability of standing waves for nonlinear Schrödinger equations with a critical rotational speed. (English) Zbl 1492.35304
Summary: We study the existence and stability of standing waves associated with the Cauchy problem for the nonlinear Schrödinger equation (NLS) with a critical rotational speed and an axially symmetric harmonic potential. This equation arises as an effective model describing the attractive Bose-Einstein condensation in a magnetic trap rotating with an angular velocity. By viewing the equation as NLS with a constant magnetic field and with (or without) a partial harmonic confinement, we establish the existence and orbital stability of prescribed mass standing waves for the equation with mass-subcritical, mass-critical, and mass-supercritical nonlinearities. Our result extends a recent work of J. Bellazzini et al. [Commun. Math. Phys. 353, No. 1, 229–251 (2017; Zbl 1367.35150)], where the existence and stability of standing waves for the supercritical NLS with a partial confinement were established.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35B35 | Stability in context of PDEs |
| 78A37 | Ion traps |
| 82C10 | Quantum dynamics and nonequilibrium statistical mechanics (general) |
Citations:
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