Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential
Reika Fukuizumi
Discrete & Continuous Dynamical Systems - A 2001, 7(3): 525-544 doi: 10.3934/dcds.2001.7.525
In this paper, we study the stability and the instability of standing waves for the nonlinear Schrödinger equation with harmonic potential. We prove the existence of stable or unstable standing waves under certain conditions on the power of nonlinearity and the frequency of wave.
keywords: harmonic potential Nonlinear Schrödinger equation standing wave orbital stability.
Stability of standing waves for a nonlinear Schrödinger equation wdelta potentialith a repulsive Dirac
Reika Fukuizumi Louis Jeanjean
Discrete & Continuous Dynamical Systems - A 2008, 21(1): 121-136 doi: 10.3934/dcds.2008.21.121
We consider a stationary nonlinear Schröodinger equation with a repulsive delta-function impurity in one space dimension. This equation admits a unique positive solution and this solution is even. We prove that it is a minimizer of the associated energy on the subspace of even functions of $H^1(\R, \C)$, but not on all $H^1(\R, \C)$, and study its orbital stability.
keywords: variational methods. Dirac delta standing waves stability nonlinear Schrödinger equation
Vortex solutions in Bose-Einstein condensation under a trapping potential varying randomly in time
Anne de Bouard Reika Fukuizumi Romain Poncet
Discrete & Continuous Dynamical Systems - B 2015, 20(9): 2793-2817 doi: 10.3934/dcdsb.2015.20.2793
The aim of this paper is to perform a theoretical and numerical study on the dynamics of vortices in Bose-Einstein condensation in the case where the trapping potential varies randomly in time. We take a deterministic vortex solution as an initial condition for the stochastically fluctuated Gross-Pitaevskii equation, and we observe the influence of the stochastic perturbation on the evolution. We theoretically prove that up to times of the order of $\epsilon^{-2}$, the solution having the same symmetry properties as the vortex decomposes into the sum of a randomly modulated vortex solution and a small remainder, and we derive the equations for the modulation parameter. In addition, we show that the first order of the remainder, as $\epsilon$ goes to zero, converges to a Gaussian process. Finally, some numerical simulations on the dynamics of the vortex solution in the presence of noise are presented.
keywords: white noise harmonic potential collective coordinates approach. stochastic partial differential equations vortices Nonlinear Schrödinger equation

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