34 research outputs found
Modulation analysis for a stochastic NLS equation arising in Bose-Einstein condensation
International audienceWe 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 ε 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 ε−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 ε 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
Representation formula for stochastic Schrödinger evolution equations and applications
International audienceWe prove a representation formula for solutions of Schrödinger 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 or in fiber optics. Our results also give a justification of diffusion-approximation for stochastic nonlinear Schrödinger equations
Instability of bound states of a nonlinear Schr\"odinger equation with a Dirac potential
We study analytically and numerically the stability of the standing waves for
a nonlinear Schr\"odinger 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 \hurad and unstable in \hu under subcritical nonlinearity.
Further we investigate the nature of instability: under critical or
supercritical nonlinear interaction, we prove the instability by blowup 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 {\em 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 Schroedinger equations with double well potential in the semiclassical limit
We consider the stationary solutions for a class of Schroedinger 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
non-linearity 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.Comment: 46 pages, 4 figure