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RandommodulationandpersistenceofsolitonsforthestochasticKdVequationA.deBouardEcolePolytechnique,Palaiseau,FranceJointworkswithA.DebusscheandE.GautierFrance-Taı¨wanjointconferenceonnonlinearPDEs,CIRM,March25-28,2008.AedBuorad
thedeterministicKdVequationtu+x3u+x(u2)=0I“Universalmodel”:Asymptoticmodelforlongwavesatthesurfaceofwater(smallamplitude,shallowwater,unidirectionalpropagation)Rigorousderivation:W.Craig,CPDE,1985Modelforplasmaphysics:Herman,J.Phys.A,1990IIntegrableequationHamiltoniansystem(action-anglevariables):infinitenumberofintegralsofmotionAllowstosolvegloballytheequation(weakly)inspacesofirregulardata:Kappeler-Toppalov,DukeMath.J.,2006 whitenoiseinvariantmeasure.AedBuorad
IInversescatteringmethods:Gardner,Green,Kruskal,Miura,PRL,1967Resolutionintosolitonsofanysmoothenoughanddecayingsolution:Ekhaus,Schuur,M2AS,1983Solitons:two-parameterfamilyofsolutionsuc0,x0(t,x)=ϕc0(xc0t+x0)withϕ(x)=3c0c02cosh2(c0x/2)IFromPDEpointofview:IEquationgloballywell-posedinHs(R),s>3/4,andHs(T),s>1/2:Bourgain;Kenig,Ponce,Vega;Colliander,Staffilani,Takaoka,TaoISolitonsareorbitallystableBenjamin,Proc.Roy.Soc.Lond.,1972andevenasymptoticallystablePego,Weinstein,CMP,1994;Martel,Merle,Arch.Rat.Mech.Anal.,01.AedBuorad
RandomperturbationsoftheKdVequationNorigorous(mathematical)derivationIForcingterm:Surfacewaves,randompressuree.g.turbulentwindonthesurface addatermξ˙(t,x)whitenoiseintimeIVariationsofthebottomtopographymodeledbyastationary(inx)randomprocess addaterm(xu)ξ˙(t)(whiteintime)Mathematically:KdVequationrewrittenintheframemovingwithvelocityξ˙:u(xξ(t))IRandompotential:uξ˙or(xu)ξ˙Plasmaphysics,Herman,J.Phys.A,1990.AedBuorad
MathematicalresultsintherandomlyperturbedcaseIAdditivecase:PDEmethods:dB,Debussche,Tsutsumi,JFA,99;Printems,JDE,99:EquationiswellposedinHs(R),s>5/8forirregularnoiseinspace(includes“localised”space-timewhitenoise)dB,Debussche,Tsutsumi,SIAM,04:SameresultforxT,butwiths>1/2,i.e.closetospace-timewhitenoise)dB,Debussche,AnnIHP,07:RandommodulationofsolitonsIntegrabilitymethods(action-anglevariables)Kuksin,Piatnitski,toappearinJMPA:Hasminski-Whithamaveragingfortu+x3uν∂x2u+x(u2)=νξ˙˙ξtwruhbiutleencnoesieniitme.A,ederBguouraldranix;moedilgnfoewka
IMultiplicativecase:PDEMethods:dB,Debussche,Int.Disc.Math.Sc.,07:ExistenceofsolutionsinL2(R)andH1(R)(fornoiseinL2orH1)Tsutsumi,preprint07:Regularnoiseinspace,u(t,x)goestozeroa.s.astgoestoinfinity,foranyu0,underspecificassumptionsonthenoise,andxTInversescatteringmethods:Garnier,J.Stat.Phys.01:Noiseoftheform(xu)ξ˙(t),(x3u)ξ˙(t)orx(u2)ξ˙(t),i.e.perturbationsofvelocity,dispersionornonlinearity;propagationofsolitons:equationsonthescatteringdata;noestimateontheremainingtermsfortheoriginalsolution.AedBuorad
OriginalmotivationoftheworkWadati,J.Phys.Soc.Japan,1983:Particularcaseofanoisethatdependsonlyontimedu+(x3u+x(u2))dt=dWwithW(t)realvaluRed,centered,Brownianmotion;thenu(t,x)=U(t,x0tW(s)ds)+W(t)withUsolutionofdeterministicKdV.IfinparticularU(t,x)=ϕc0(xc0t),thentZZE(u(t,x))=E(ϕc0(xc0tW(s)ds))=ϕc0(xc0ty)µ(dy)R0Rwithµ=L(0tW(s)ds)∼N(0,t3/3);oneeasilygetsmaxE(u(t,x))Ct3/2RxSameresult,butwithCt1/2ifnoise(xu)dW/dtIsitpossibletogetsuchresultsformoregeneralnoises(dependingonspace)with,e.g.,amplitudegoestozero?.AedBuorad
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