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Article Dans Une Revue Discrete and Continuous Dynamical Systems - Series A Année : 2010

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

Résumé

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.
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Dates et versions

hal-00216208 , version 1 (24-01-2008)

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Anne de Bouard, Eric Gautier. Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise. Discrete and Continuous Dynamical Systems - Series A, 2010, 26 (3), pp.857 - 871. ⟨10.3934/dcds.2010.26.857⟩. ⟨hal-00216208⟩
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