In this paper, we study the stability of standing waves for the nonlinear Schrödinger equation on the unit ball in R N with Dirichlet boundary condition. We generalize the result of Fibich and Merle (2001 Physica D 155 132-58), which proves the orbital stability of the least-energy solution with the cubic power nonlinearity in two space dimension. We also obtain several results concerning the excited states in one space dimension. Specifically, we show the linear stability of the first three excited states and we give a proof of the orbital stability of the kth excited state, restricting ourselves to the perturbation of the same symmetry as the kth excited state. Finally, our numerical simulations on the stability of the kth excited state are presented.

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In this paper, we study the structure of radially symmetric standing waves for the nonlinear Schrödinger equation with harmonic potential, which arises in a wide variety of applications and is known as the Gross-Pitaevskii equation in the context of Bose-Einstein condensates with parabolic traps. Both global and local bifurcation behaviour are determined showing the existence of infinitely symmetric localized states. In particular, our theory provides a theoretical proof of the existence of a solution with prescribed numbers of zeros depending on the frequency of the wave. After a few remarks concerning the critical case, numerical computations are finally presented in order to provide an illustration of the theoretical results that have been obtained and also to investigate the supercritical case for which only few results are known.

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