Searcharxiv⌕ Search

arXiv subjects

Eric Gautier

Publications and source records attributed to Eric Gautier.

25 records · Page 2Linked to original sources
arXiv2008

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.

math.PR↗
arXiv2008

Small noise asymptotic of the timing jitter in soliton transmission

We consider the problem of the error in soliton transmission in long-haul optical fibers caused by the spontaneous emission of noise inherent to amplification. We study two types of noises driving the stochastic focusing cubic one dimensional nonlinear Schrödinger equation which appears in physics in that context. We focus on the fluctuations of the mass and arrival time or timing jitter. We give the small noise asymptotic of the tails of these two quantities for the two types of noises. We are then able to prove several results from physics among which the Gordon--Haus effect which states that the fluctuation of the arrival time is a much more limiting factor than the fluctuation of the mass. The physical results had been obtained with arguments difficult to fully justify mathematically.

math.PR↗
arXiv2006

Stochastic nonlinear Schrodinger equations driven by a fractional noise - Well posedness, large deviations and support

We consider stochastic nonlinear Schrodinger equations driven by an additive noise. The noise is fractional in time with Hurst parameter H in (0,1). It is also colored in space and the space correlation operator is assumed to be nuclear. We study the local well-posedness of the equation. Under adequate assumptions on the initial data, the space correlations of the noise and for some saturated nonlinearities, we prove a sample path large deviations principle and a support result. These results are stated in a space of exploding paths which are Holder continuous in time until blow-up. We treat the case of Kerr nonlinearities when H > 1/2.

math.PR↗
arXiv2006

Exit from a basin of attraction for stochastic weakly damped nonlinear Schrödinger equations

We consider weakly damped nonlinear Schrödinger equations perturbed by a noise of small amplitude. The small noise is either complex and of additive type or real and of multiplicative type. It is white in time and colored in space. Zero is an asymptotically stable equilibrium point of the deterministic equations. We study the exit from a neighborhood of zero, invariant by the flow of the deterministic equation, in $\xLtwo$ or in $\xHone$. Due to noise, large fluctuations off zero occur. Thus, on a sufficiently large time scale, exit from these domains of attraction occur. A formal characterization of the small noise asymptotic of both the first exit times and the exit points is given.

math.NA↗
arXiv2004

Uniform large deviations for the nonlinear Schrodinger equation with multiplicative noise

Uniform large deviations for the laws of the paths of the solutions of the stochastic nonlinear Schrodinger equation when the noise converges to zero are presented. The noise is a real multiplicative Gaussian noise. It is white in time and colored in space. The path space considered allows blow-up and is endowed with a topology analogue to a projective limit topology. Thus a large variety of large deviation principle may be deduced by contraction. As a consequence, asymptotics of the tails of the law of the blow-up time when the noise converges to zero are obtained.

math.AP↗
arXiv2004

Large deviations and support results for nonlinear Schrodinger equations with additive noise and applications

Sample path large deviations for the laws of the solutions of stochastic nonlinear Schrodinger equations when the noise converges to zero are presented. The noise is a complex additive gaussian noise. It is white in time and colored space wise. The solutions may be global or blow-up in finite time, the two cases are distinguished. The results are stated in trajectory spaces endowed with projective limit topologies. In this setting, the support of the law of the solution is also characterized. As a consequence, results on the law of the blow-up time and asymptotics when the noise converges to zero are obtained. An application to the transmission of solitary waves in fiber optics is also given.

math.AP↗