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Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential. (English) Zbl 0992.35094

The existence and stability of stationary states in the nonlinear Schrödinger equation in spaces of different dimension and with different nonlinearity powers are considered, in the presence of an external trapping potential. The most important application of this problem is to the description of Bose-Einstein condensates, in which case the nonlinearity is cubic. Formal proofs of the existence of stationary states are given, as well as formal stability criteria for different values of the space dimension and nonlinearity power.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B35 Stability in context of PDEs
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
35A15 Variational methods applied to PDEs
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