September  2010, 26(3): 857-871. doi: 10.3934/dcds.2010.26.857

Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise

1. 

Centre de Mathématiques Appliquées, UMR 7641 CNRS/Ecole Polytechnique, 91128 Palaiseau cedex, France

2. 

ENSAE-CREST, 3 avenue Pierre Larousse, 92245 Malakoff Cedex, France

Received  April 2008 Revised  August 2009 Published  December 2009

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 $\epsilon$. The initial datum gives rise to a soliton when $\epsilon=0$. 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 $\epsilon^{-2}$. 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 $\epsilon$ 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.
Citation: Anne de Bouard, Eric Gautier. Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 857-871. doi: 10.3934/dcds.2010.26.857
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