Reika Fukuizumi

Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

We study analytically and numerically the stability of the standing waves for a nonlinear Schrodinger equation with a point defect and a power type nonlinearity. A main difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves, and it is overcome by a perturbation method and continuation arguments. Among others, in the case of a repulsive defect, we show that the standing wave solution is stable in H^1_rad and unstable in H^1 under subcritical nonlinearity. Further we investigate the nature of instability: under critical or supercritical nonlinear interaction, we prove the instability by blow-up in the repulsive case by showing a virial theorem and using a minimization method involving two constraints. In the subcritical radial case, unstable bound states cannot collapse, but rather narrow down until they reach the stable regime (a finite-width instability). In the non-radial repulsive case, all bound states are unstable, and the instability is manifested by a lateral drift away from the defect, sometimes in combination with a finite-width instability or a blowup instability.

Bifurcation and Stability for Nonlinear Schrödinger Equations with Double Well Potential in the Semiclassical Limit

We consider the stationary solutions for a class of Schrödinger equations with a symmetric double-well potential and a nonlinear perturbation. Here, in the semiclassical limit we prove that the reduction to a finite-mode approximation give the stationary solutions, up to an exponentially small term, and that symmetry-breaking bifurcation occurs at a given value for the strength of the nonlinear term. The kind of bifurcation picture only depends on the nonlinearity power. We then discuss the stability/instability properties of each branch of the stationary solutions. Finally, we consider an explicit one-dimensional toy model where the double well potential is given by means of a couple of attractive Dirac’s delta pointwise interactions.

Representation formula for stochastic Schrödinger evolution equations and applications

We prove a representation formula for solutions of Schrodinger equations with potentials multiplied by a temporal real-valued white noise in the Stratonovich sense. Using this formula, we obtain a dispersive estimate which allows us to study the Cauchy problem in L2 or in the energy space of model equations arising in Bose–Einstein condensation, Abdullaev et al (2001 Nonlinearity and Disorder: Theory and Applications (NATO Science Series vol 45) ed F Abdullaev et al (Dodrecht: Kluwer)), or in fiber optics, Abdullaev et al (2000 Physica D 135 369–86). Our results also give a justification of diffusion-approximation for stochastic nonlinear Schrodinger equations.

Stationary problem related to the nonlinear Schrödinger equation on the unit ball

In this paper, we study the stability of standing waves for the nonlinear Schrodinger equation on the unit ball in 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.

Modulation analysis for a stochastic NLS equation arising in Bose–Einstein condensation

We study the asymptotic behavior of the solution of a model equation for Bose-Einstein condensation, in the case where the trapping potential varies randomly in time. The model is the so called Gross-Pitaevskii equation, with a quadratic potential with white noise fluctuations in time whose amplitude e tends to zero. The initial condition of the solution is a standing wave solution of the unperturbed equation. We prove that up to times of the order of e −2 , the solution decomposes into the sum of a randomly modulated standing wave and a small remainder, and we derive the equations for the modulation parameters. In addition, we show that the first order of the remainder, as e goes to zero, converges to a Gaussian process, whose expected mode amplitudes concentrate on the third eigenmode generated by the Hermite functions, on a certain time scale.