N-soliton solutions to the DKP equation and Weyl g

arXiv:nlin/0602031v1[nlin.SI]14Feb2006N-SOLITONSOLUTIONSTOTHEDKPEQUATIONANDWEYLGROUPACTIONSYUJIKODAMA∗ANDKEN-ICHIMARUNO†Abstract.WestudysolitonsolutionstotheDKPequationwhichisdefinedbytheHirotabilinearform,(−4DxDt+D4x+3D2y)τn·τn=24τn−1τn+1,(2Dt+D3x∓3DxDy)τn±1·τn=0n=1,2,....whereτ0=1.Theτ-functionsτnaregivenbythepfaffiansofcertainskew-symmetricmatrix.Weidentifyone-solitonsolutionasanelementoftheWeylgroupofD-type,anddiscussageneralstructureoftheinteractionpatternsamongthesolitons.Solitonsolutionsarecharacterizedby4N×4Nskew-symmetricconstantmatrixwhichwecalltheB-matrices.WethenfindthatonecanhaveM-solitonsolutionswithMbeinganynumberfromNto2N−1forsomeofthe4N×4NB-matriceshavingonly2Nnonzeroentriesintheuppertriangularpart(thenumberofsolitonsobtainedfromthoseB-matriceswaspreviouslyexpectedtobejustN).1.IntroductionWestudysolitonsolutionsoftheDKPequation[1,3,9],(1.1)(−4DxDt+D4x+3D2y)τn·τn=24τn−1τn+1,(2Dt+D3x∓3DxDy)τn±1·τn=0n=1,2...,whereτ0=1,andDx,DyandDtaretheusualHirotaderivatives.Foreachn,thevariablesu=2(lnτn)xxandv±=τn±1/τndefinethecoupledKPequation(seeforexample[3]),(−4ut+uxxx+6uux)x+3uyy=24(v+v−)xx,2v±t+v±xxx+3uv±x∓3(v±xy+v±Rxuydx)=0.Thissetofequationsadmitsaclassofparticularsolutions,calledsolitonsolutionssimilartothoseoftheKPequation(seeforexample[5]),andinthispaperwedescribesomepropertiesofthosesolutions.Remark1.1.TheDKPequationwasgivenasafirstmemberoftheDKPhierarchyin[6].In[3],thecoupledKPequationwasintroducedasanextensionoftheKPequationwhosesolutionsaregivenbythepfaffianform.TheDKPequationwasrediscoveredasthePfafflatticedescribingthepartitionfunctionofaskew-symmetricmatrixmodelin[1,7],andwasalsofoundasanorbitofsomeinfinite-dimensionalCliffordgroupactionin[9].Theτnfunctionsaregivenbythepfaffiansof2n×2nskew-symmetricmatricesQnwhoseentriesaredenotedbyQi,jfor1≤ij≤2nwithQj,i=−Qi,j,(1.2)τn=Pf(Qn)=X1=i1...in2nikjk,k=1,...,nσ(i1,j1,...,in,jn)Qi1,j1Qi2,j2···Qin,jn∗PartiallysupportedbyNSFgrantDMS0404931.†SupportedbyCOEprogramatFacultyofMathematics,KyushuUniversity.12YUJIKODAMA∗ANDKEN-ICHIMARUNO†Eachcoefficientσ(i1,j1,...,in,jn)givesasigncorrespondingtotheparityofthepermutation,σ:=sign12···2n−12ni1j1···injnHeretheelementsQi,jsatisfy∂∂tkQi,j=Qi+k,j+Qi,j+k,k=1,2,...wheret1=x,t2=y,t3=tandothersarethesymmetryparameters.ArealizationofQi,jisgivenby[3],Qi,j=φ(i−1)φ(j−1)ψ(i−1)ψ(j−1),forijwhereφ(k)=∂kφ/∂xk(thesameforψ(k))andthefunctionsφandψsatisfythesameequationsforxandtevolutions,∂φ∂y=∂2φ∂x2,∂φ∂t=∂3φ∂x3.Foranexampleoffinitedimensionalsolution,weconsider(1.3)φ(x,y,t)=MXm=1amEm(x,y,t),ψ(x,y,t)=MXm=1bmEm(x,y,t),withsomeconstantsam,bmform=1,...,M.ThefunctionEm(x,y,t)istheexponentialfunction,Em(x,y,t):=eθmwithθm(x,y,t)=pmx+p2my+p3mt+θ0m,wherepmandθ0marearbitraryconstants,andthroughoutthispaperweassumetheparametersp:=(p1,p2,...,pM)tobeorderedas,(1.4)p1p2···pM.Withthoseφandψ,theelementsQi,j,1≤ij≤2nbecome(1.5)Qi,j=X1≤kl≤Mbk,lE(i−1)kE(j−1)kE(i−1)lE(j−1)l=X1≤kl≤Mbk,l(pkpl)i−1(pj−il−pj−ik)Ek,l,wherebk,l=akbl−albk,andEk,l:=EkEl=exp(θk+θl).AsageneralizationofthisformofQi,j,weconsideranarbitraryM×Mskew-symmetricmatrixB=(bk,l)1≤k,l≤M.Wethennotethatthe2n×2nmatrixQnintheτ-functionτncanbeexpressedas(1.6)Qn(x,y,t)=En(x,y,t)BEn(x,y,t)TwhereEnisa2n×MmatrixwithitstransposeETn,anditisgivenbyEn=E1E2···EM............E(2n−1)1E(2n−1)2···E(2n−1)M.Inthispaper,wediscussaclassificationproblemofseveralsolitonsolutionsgivenby(1.6)intermsoftheB-matrix.NoteherethatM2forn≥1,sincethecasewithM=2givesatrivialsolution,i.e.τ1=Q1,2=b1,2(p2−p1)E1E2andu=2(lnτ1)xx=0.AlsonotethatoneneedsM2nforu=2(lnτn)xx6=0.Oneshouldalsonotethatifτn+1=0(i.e.v+=0),thecorrespondingsolutionsatisfiestheKPequation.ThisconditioncanbeofcourseobtainedfromthestructureoftheB-matrix.Forexample,ifwetakeM=3,thenτ2vanishesidentically(thesizeofthepfaffianis4×4,buttheindependentexponentialsarethreeorless).Thisimpliesthatτ1withM=3givesasolutionoftheN-SOLITONSOLUTIONSTOTHEDKPEQUATIONANDWEYLGROUPACTIONS3KPequation,anditgiveseitheroneKPsolitonsolutionoraresonantY-shapeKPsoliton:Witha3×3B-matrix,wehaveτ1=Q1,2=b1,2(p2−p1)E1E2+b1,3(p3−p1)E1E3+b2,3(p3−p2)E2E3.Thefunctionu=2(lnτ1)xxgivesoneKPsolitonsolutionifoneofbi,jiszero(withothersbeingpositive),andY-shapeKPsolitonifallbi,jarepositive.Forexample,withb1,2=0,wehaveτ1=(b1,3(p3−p1)E1+b2,3(p3−p2)E2)E3whichleadsto(1.7)w(x,y,t):=∂∂xlnτ1=p3+12(p1+p2)+12(p2−p1)tanh12(θ2−θ1).Theasymptoticvaluesofwthentakew(x,y,t)→p1+p3,forx→−∞p2+p3,forx→∞Noteherethattheonesolitonexchangetheasymptoticvaluesof(1,3):=p1+p3and(2,3):=p2+p3,thatis,thisonesolitonpermutesthenumbers,(1)↔(2).WethenlabelonesolitonoftheKPequationasanelementofthepermutationgroupW,inthiscaseW=S3,thesymmetrygroupoforder3.Wedenotetheonesolitonsolution(1.7)by[1:2],andcallthistypeofsolitonA-soliton(“A”standsfortypeALiealgebrawhichistheunderlyingsymmetryalgebraforKPequation).Ingeneral,wedenoteoneA-solitonby[i:j],ifthefunctionwexchangespi↔pjwithpipj.Wesometimeidentify[i:j]asanelementofthesymmetrygroup.AgenericsolutionoftheDKPequationisthenobtainedforM≥4.Inparticular,weobtainonesolitonsolutio