05 - The Korteweg-de Vries equation with multiplicative noise : existence of solutions and random modulation of solitons - Arnaud DEBUSSCHE

In this work, we consider the Korteweg- de Vries equation perturbed by a random force of white noise type, additive or multiplicative. In a series of work, in collaboration with Y. Tsutsumi, we have studied existence and uniqueness in the additive case for very irregular noises. These use the functional framework introduced by J. Bourgain. We use similar tools to prove existence and uniqueness for a multiplicative noise. We are not able to consider irregular noises and have to assume that the driving Wiener process has paths in or . However, contrary to the additive case, we are able to treat spatially homogeneous noises. Then, we try to understand the effect of a small noise with amplitude on the propagation of a soliton. We prove that, on a time scale proportional to , a solution initially equal to the soliton but perturbed by a noise of the type above remains close to a soliton with modulated speed and position. The modulated speed and position are semi-martingales and we write the stochastic equations they satisfy. We prove also that a Central Limit Theorem holds so that, on the time scale described above, the solutions can formally be written as the sum of the modulated soliton and a gaussian remainder term of order . In the multiplicative case, we can go further. We prove that the gaussian part converges in distribution to a stationary process. Also, the equations for the modulation parameters allow to give a justification for the phenomenon called "soliton diffusion" observed in numerical simulations: the averaged soliton decays like . We obtain . This a joint work with A. de Bouard. Arnaud DEBUSSCHE. ENS Cachan. Document associé : support de présentation : http://epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw?CODE_FICHIER=1182789894119 (pdf) Bande son disponible au format mp3 Durée : 45 mn