Random modulation and persistence of solitons for the stochastic KdV equation

A. Deboüard

Description

Niveau: Supérieur
Random modulation and persistence of solitons for the stochastic KdV equation A. de Bouard Ecole Polytechnique, Palaiseau, France Joint works with A. Debussche and E. Gautier France-Taıwan joint conference on nonlinear PDEs, CIRM, March 25-28, 2008 A. de Bouard

• white noise invariant


• bottom topography modeled


• pde methods


• ?˙ white


• surface waves


• velocity ?˙


• space-time white


• noise

Sujets

• École polytechnique
• Surface wave

Informations

• Publié par: chaeh
• Nombre de lectures: 9
• Langue: English

Extrait

Randommodulationandpersistenceofsolitons forthestochasticKdVequation A.deBouard EcolePolytechnique,Palaiseau,France JointworkswithA.DebusscheandE.Gautier France-Taı¨wanjointconferenceonnonlinearPDEs,CIRM, March25-28,2008 .AedBuorad

thedeterministicKdVequation ∂ t u+∂ x 3 u+∂ x (u 2 )=0 I “Universalmodel”: Asymptoticmodelforlongwavesatthesurfaceofwater (smallamplitude,shallowwater,unidirectionalpropagation) Rigorousderivation:W.Craig,CPDE,1985 Modelforplasmaphysics:Herman,J.Phys.A,1990 I Integrableequation Hamiltoniansystem(action-anglevariables):infinitenumber ofintegralsofmotion Allowstosolvegloballytheequation(weakly)inspacesof irregulardata:Kappeler-Toppalov,DukeMath.J.,2006  whitenoiseinvariantmeasure .AedBuorad

I Inversescatteringmethods:Gardner,Green,Kruskal, Miura,PRL,1967 Resolutionintosolitonsofanysmoothenoughanddecaying solution:Ekhaus,Schuur,M2AS,1983 Solitons:two-parameterfamilyofsolutions u c 0 ,x 0 (t,x)=ϕ c 0 (x − c 0 t+x 0 )with ϕ(x)= 3c 0 c 0 2cosh 2 ( √ c 0 x/2) I FromPDEpointofview: IEquationgloballywell-posedinH s ( R ),s> − 3/4,and H s ( T ),s> − 1/2:Bourgain;Kenig,Ponce,Vega;Colliander, Staffilani,Takaoka,Tao ISolitonsareorbitallystableBenjamin,Proc.Roy.Soc. Lond.,1972andevenasymptoticallystablePego,Weinstein, CMP,1994;Martel,Merle,Arch.Rat.Mech.Anal.,01 .AedBuorad