547 research outputs found
Noise Prevents Singularities in Linear Transport Equations
A stochastic linear transport equation with multiplicative noise is
considered and the question of no-blow-up is investigated. The drift is assumed
only integrable to a certain power. Opposite to the deterministic case where
smooth initial conditions may develop discontinuities, we prove that a certain
Sobolev degree of regularity is maintained, which implies H\"older continuity
of solutions. The proof is based on a careful analysis of the associated
stochastic flow of characteristics
Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise
We consider two exit problems for the Korteweg-de Vries equation perturbed by
an additive white in time and colored in space noise of amplitude a. The
initial datum gives rise to a soliton when a=0. It has been proved recently
that the solution remains in a neighborhood of a randomly modulated soliton for
times at least of the order of a^{-2}. We prove exponential upper and lower
bounds for the small noise limit of the probability that the exit time from a
neighborhood of this randomly modulated soliton is less than T, of the same
order in a and T. We obtain that the time scale is exactly the right one. We
also study the similar probability for the exit from a neighborhood of the
deterministic soliton solution. We are able to quantify the gain of eliminating
the secular modes to better describe the persistence of the soliton
Blow-up for the stochastic nonlinear Schrodinger equation with multiplicative noise
We study the influence of a multiplicative Gaussian noise, white in time and
correlated in space, on the blow-up phenomenon in the supercritical nonlinear
Schrodinger equation. We prove that any sufficiently regular and localized
deterministic initial data gives rise to a solution which blows up in
arbitrarily small time with a positive probability.Comment: Published at http://dx.doi.org/10.1214/009117904000000964 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Long wave expansions for water waves over random topography
In this paper, we study the motion of the free surface of a body of fluid
over a variable bottom, in a long wave asymptotic regime. We assume that the
bottom of the fluid region can be described by a stationary random process
whose variations take place on short length scales and which
are decorrelated on the length scale of the long waves. This is a question of
homogenization theory in the scaling regime for the Boussinesq and KdV
equations. The analysis is performed from the point of view of perturbation
theory for Hamiltonian PDEs with a small parameter, in the context of which we
perform a careful analysis of the distributional convergence of stationary
mixing random processes. We show in particular that the problem does not fully
homogenize, and that the random effects are as important as dispersive and
nonlinear phenomena in the scaling regime that is studied. Our principal result
is the derivation of effective equations for surface water waves in the long
wave small amplitude regime, and a consistency analysis of these equations,
which are not necessarily Hamiltonian PDEs. In this analysis we compute the
effects of random modulation of solutions, and give an explicit expression for
the scattered component of the solution due to waves interacting with the
random bottom. We show that the resulting influence of the random topography is
expressed in terms of a canonical process, which is equivalent to a white noise
through Donsker's invariance principle, with one free parameter being the
variance of the random process . This work is a reappraisal of the paper
by Rosales & Papanicolaou \cite{RP83} and its extension to general stationary
mixing processes
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
Wave energy localization by self-focusing in large molecular structures: a damped stochastic discrete nonlinear Schroedinger equation model
Wave self-focusing in molecular systems subject to thermal effects, such as
thin molecular films and long biomolecules, can be modeled by stochastic
versions of the Discrete Self-Trapping equation of Eilbeck, Lomdahl and Scott,
and this can be approximated by continuum limits in the form of stochastic
nonlinear Schroedinger equations.
Previous studies directed at the SNLS approximations have indicated that the
self-focusing of wave energy to highly localized states can be inhibited by
phase noise (modeling thermal effects) and can be restored by phase damping
(modeling heat radiation).
We show that the continuum limit is probably ill-posed in the presence of
spatially uncorrelated noise, at least with little or no damping, so that
discrete models need to be addressed directly. Also, as has been noted by other
authors, omission of damping produces highly unphysical results.
Numerical results are presented for the first time for the discrete models
including the highly nonlinear damping term, and new numerical methods are
introduced for this purpose. Previous conjectures are in general confirmed, and
the damping is shown to strongly stabilize the highly localized states of the
discrete models. It appears that the previously noted inhibition of nonlinear
wave phenomena by noise is an artifact of modeling that includes the effects of
heat, but not of heat loss.Comment: 22 pages, 13 figures, revision of talk at FPU+50 conference in Rouen,
June 200
Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion
International audienceThis article is devoted to the numerical study of a nonlinear Schrödinger equation in which the coefficient in front of the group velocity dispersion is multiplied by a real valued Gaussian white noise. We first perform the numerical analysis of a semi-discrete Crank-Nicolson scheme in the case when the continuous equation possesses a unique global solution. We prove that the strong order of convergence in probability is equal to one in this case. In a second step, we numerically investigate, in space dimension one, the behavior of the solutions of the equation for different power nonlinearities, corresponding to subcritical, critical or supercritical nonlinearities in the deterministic case. Numerical evidence of a change in the critical power due to the presence of the noise is pointed out
A semi-discrete scheme for the stochastic Landau-Lifshitz equation
We propose a new convergent time semi-discrete scheme for the stochastic Landau-Lifshitz-Gilbert equation. The scheme is only linearly implicit and does not require the resolution of a nonlinear problem at each time step. Using a martingale approach, we prove the convergence in law of the scheme up to a subsequence
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