On variational formulations for steady water waves

OnvariationalformulationsforsteadywaterwavesMarkD.GrovesJohnF.TolandSchoolofMathematicalSciences,UniversityofBath,ClavertonDown,Bath,BA27AY,England.SupportedbyanE.P.S.R.CResearchFellowship.Presentaddress:DepartmentofMathematicalSciences,LoughboroughUniversityofTechnology,Loughborough,Leicestershire,LE113TU,UK11Introduction1.1BackgroundandrecentdevelopmentsThewater-waveproblemisthestudyoftwo-dimensional,inviscid,irrotationalfluidflowinadomainoffiniteorinfinitedepth,boundedabovebyafreesurfaceanddrivenbytheeffectsofgravityandpossiblysurfacetension.Thefirstcontributiontothestudyofvariationalformula-tionsofthisproblemwasmadebyLuke(1967),whopublishedaformalvariationalprinciplethatrecoversthewater-waveequations.AHamiltonianformulationoftheproblemwasreportedbyZakharov(1968),andwaspursuedindependentlybyBroer(1974;1975),Broer,vanGroesen&Timmers(1976)andMiles(1977)(seealsoMilder(1977)).Sincethentherehasbeenagreatdealofactivityconcerningvariationalformulationsofexactandmodelwater-waveequations,notablybyMiles(1981)(animportantsummaryofearlydevelopmentsinHamiltonianwater-wavethe-ory),Benjamin&Olver(1982)(whousedtheHamiltonianstructuretoclassifythesymmetriesandconservationlawsofthewater-waveproblem),Benjamin(1984)(atreatiseonvariationalprinciplesforHamiltoniansystemsandinparticular(x6.1)analternativeHamiltonianformula-tionforwaterwaves)andRadder(1992)(afurtherHamiltonianformulationforwaterwaves).Thepastfewyearshaveseenmuchinterestinvariationalformulationsofthetwo-dimensionalsteadywater-waveprobleminwhichthefree-surfaceflowsarestationaryrelativetoauniformly-translatingframeofreference.Themostwell-knownexamplesareperiodic-andsolitary-waveflows,whichhavebeenthesubjectofextensiveresearchsincethepioneeringworkofStokesandLordRayleighinthemid-nineteenthcentury(seeLamb(1924,xx250–255)andWhitham(1974,x13.12–13.13)forathoroughdiscussionoftheworkoftheseauthors).Inanewapproachtothequestionoftheexistenceoftravellingwavesofsmallamplitude,Kirchg¨assner(1988)hasre-centlyintroducedapowerfulmethodofreducingthewater-waveproblemtoanequivalentfinite-dimensionalsystemofordinarydifferentialequations.Theanalysisofthereducedproblemhasleadtoimportantresults,inparticulartothefirstexistencetheoremsforsolitarywavesinthepres-enceofsurfacetension(Amick&Kirchg¨assner1989;Iooss&Kirchg¨assner1990).OnemayregardKirchg¨assner’smethodasacentre-manifoldreductionofaspecialkindwhichinvolveswritingasysteminwhich,strictlyspeaking,thereisnotime-likevariable,astheanalogyofady-namicalsystem.Mielke(1991)hasdiscussedcentre-manifoldreductionsofdynamicalsystemsingeneralandhasshownthatifthefullsystemhasaHamiltonianstructurethensodoesthereducedsystem.Asaconsequenceofthiswork,Mielke(1991,Chapter9),Baesens&MacKay(1992)andBridges(1992;1994)havestudiedHamiltonianformulationsofthesteadywater-waveproblemwrittenasadynamicalsystem.Mielkediscussesthecaseoffinitedepthandnon-zerosurfaceten-2sion,whileBaesens&MacKaytreattheproblemwithinfinitedepthandzeroornon-zerosurfacetension.AlthoughtheequivalencebetweentheexactEulerequationsforaninviscid,irrotationalsteadyflowwithafreeboundaryandthereducedsystemisvalidonlyforsolutionsofsmall-amplitude,thecentre-manifoldapproachmotivatedBaesens&MacKaytomakesomeimportantobservationsconcerningthebifurcationsoflarge-amplitudeperiodicStokeswaves,previouslydiscoverednumericallybyChenandSaffman(1980),baseduponanalogieswithbifurcationthe-oryforfinite-dimensionalHamiltoniansystems.BridgeswasconcernedwithgivingageneralaccountoftheHamiltonianformalismforsteadytravellingwavesbaseduponphysicalconsider-ationsandwithoutrecoursetospecificdifferentiablemanifolds.Hisdiscussion,whichextendsandgeneralisesthatofBenjamin(1972,Appendix;1984,x6)includesanaturalinterpretationofthesymplecticformandHamiltonianintermsoffamiliarphysicalquantities.ThepresentcontributionconcernsthemathematicalframeworkoftheBaesens-MacKayandMielketheories.Itseekstohighlightcertainnovelaspectsoftheaboveproblemsandtolaythefoundationforfurtherwork(Buffoni,Groves&Toland1994).1.2Thesteadywater-waveproblemTheclassicaltime-dependentproblemforsurfacewavesonatwo-dimensionalexpanseofwaterwithundisturbeddepthh1isformulatedasfollows.Let(x;y)denotetheusualCartesiancoordinates.ThefluidoccupiesthedomainD=f(x;y):x2R;y2(0;(x;t))g,where0isafunctionofthespatialcoordinatexandtimetthatisequaltohwhenthefluidisatrest.IntermsofanEulerianvelocitypotential(x;y;t),themathematicalproblemistosolveLaplace’sequationxx+yy=0inD(1)withboundaryconditionsy=0ony=0;(2)t=yxxony=(x;t);(3)t=122x122yg(h)+xp1+2x#xony=(x;t);(4)inwhich0isthecoefficientofsurfacetension(e.g.seeBenjamin(1974),Lamb(1924,ch.IX),Whitham(1974,ch.13)).Equation(2)isthekinematicconditionthatwatercannotpermeatetherigidhorizontalboundaryaty=0,while(3),(4)arerespectivelythekinematicanddynamicconditionsatthefreesurface.3Wavesthataresteadywithrespecttoauniformly-translatingframeofreferencearedescribedbysolutionsofthespecialform(x;t)=(xct);(x;y;t)=(xct;y).Substitutingthisformof;into(1),(2),(3),(4)andusinganewperturbedvelocitypotential~=cx,onefinds