a r X i v :0710.4015v 1 [n l i n .S I ] 22 O c t 2007A new multi-component KP hierarchyXiaojun Liu †,Yunbo Zeng ‡∗,Runliang Lin ‡†Department of Applied Mathematics,China Agricultural University,Beijing 100083,P.R.China‡Department of Mathematical Sciences,Tsinghua University,Beijing 100084,P.R.ChinaAbstractA method is proposed to construct a new multi-component KP hierarchy,which includes two types of KP equation with self-consistent sources and admits reductions to k -constrained KP hierarchy and to Gelfand-Dickey hierarchy with sources.It provides a general way to construct soliton equations with sources and their Lax representations.PACS:02.30.IkKeywords:multi-component KP hierarchy;KP equation with self-consistent sources;Gelfand-Dickey hierarchy with self-consistent sources;k -constrained KP hierarchy;Lax representation1IntroductionMulti-component generalizations of KP hierarchy attract a lot of interests from both physical and mathematical points of view [1–7].The multi-component KP (mcKP)hierarchy given in [1]contains many physically relevant nonlinear integrable systems,such as Davey-Stewartsonequation,two-dimensional Toda lattice and three-wave resonant interaction ones.There ex-ist several equivalent formulations of this mcKP hierarchy:matrix pseudo-differential op-erator(Sato)formulation,τ-function approach via matrix Hirota bilinear identities,multi-component free fermion formulation.A coupled KP hierarchy was generated through the procedure of so-called Pfaffianization[8].It was shown in[9]that this coupled KP hierar-chy can be reformulated as a reduced case of the2-component KP hierarchy.Another kind of multi-component KP equation is the so-called KP equation with self-consistent sources, which was initiated by V.K.Mel’nikov[10–12].For example,thefirst type of KP equation with self-consistent sources(KPSCS)reads[10,11,13](4u t−12uu x−u xxx)x−3u yy+4Ni=1(q i r i)xx=0,(1a)q i,y=q i,xx+2uq i,i=1,...,N,(1b)r i,y=−r i,xx−2ur i.(1c) The second type of KPSCS is[10,14]4u t−12uu x−u xxx−3D−1u yy=3Ni=1[q i,xx r i−q i r i,xx+(q i r i)y],(2a)q i,t=q i,xxx+3uq i,x+32q iNj=1q j r j+32r i D−1u y−32u x r i,(2c)where D−1stands for the inverse of ddx.The notation u′i=u i,x is used in this paper.The commutativity of∂t nflows give rise to the zero-curvature equations of KP hierarchyB n,tk −B k,tn+[B n,B k]=0.In this paper,wefirst introduce a new vectorfield∂τkwhich is a linear combination of all vectorfields∂t n.Then we introduce a new Lax type equation which consist of theτk-flow and the evolutions of wave functions.Under the evolutions of wave functions,the commutativityof∂τk -flow and∂tk-flows gives rise to a new multi-component KP hierarchy.This hierarchyenables us to obtain thefirst and second types of KPSCS(i.e.,(1)and(2))in a different way from those in[10–14,16]and to get their Lax representations directly.This implies that the new mcKP hierarchy obtained in this paper is different from the mcKP hierarchy given in[1]. Moreover,this new mcKP hierarchy can be reduced to two integrable hierarchies,i.e.,the Gelfand-Dickey hierarchy with self-consistent source(GDHWS)[17]and the k-constrained KP hierarchy(k-KPH)[18,19].The GDHWS includes thefirst type of KdV equation with self-consistent sources and thefirst type of Boussinesq equation with self-consistent sources. While,the k-KPH includes the second type of KdV equation with self-consistent sources(the Yajima-Oikawa equation)and the second type of Boussinesq equation with self-consistent sources.Thus,the method proposed in this paper to construct the new mcKP hierarchy provides a general way tofind soliton equation with self-consistent sources as well as their Lax representations.Our paper will be organized as follows.In section2,we construct the new mcKP hierarchy and show that it contains thefirst and second types of KPSCS.In section3,the new mcKP hierarchy is reduced to the Gelfand-Dickey hierarchy with self-consistent source and the k-constrained KP hierarchy.In section4,some conclusions are given.2New multi-component KP hierarchyWith the help of formal dressing method,the L operator for KP hierarchy can be written in the following form[15]L=φ∂φ−1,φ=1+w1∂−1+w2∂−2+ (4)And the evolution of the dressing operatorφis given byφtn=−L n−φ.(5)The wave function and the adjoint one are then given byω(t,z)=φexp(ξ(t,z)),ω∗(t,z)=(φ∗)−1exp(−ξ(t,z)), whereξ(t,z)= i>0t i z i.They satisfy the following equationsLw(t,z)=zw(t,z),∂∂t nw∗(t,z)=−B∗n(w∗(t,z)).(6b) It was proved in[15]that the principle part of the resolvent defined byT(z)−= i∈Z L i−z−i−1,(7a) can be written asT(z)−=w(t,z)∂−1w∗(t,z).(7b) For anyfixed k∈N,we define a new variableτk whose vectorfield is∂τk =∂tk−Ni=1 s≥0ζ−s−1i∂t s,whereζi’s are arbitrary distinct non-zero parameters.Theτk-flow is given byLτk =∂tkL−Ni=1 s≥1ζ−s−1i∂t s L=[B k,L]−N i=1 s≥0ζ−s−1i[B s,L] =[B k,L]+Ni=1 s∈Nζ−s−1i[L s−,L].Define˜B k by˜B k =B k+Ni=1 s∈Zζ−s−1i L s−,which,according to(7),can be written as˜B k =B k+Ni=1w(t,ζi)∂−1w∗(t,ζi).By setting q i=w(t,ζi),r i=w∗(t,ζi),we have˜B k =B k+Ni=1q i∂−1r i,(8a)where q i and r i satisfy the following equationsq i,tn=B n(q i),r i,t n=−B∗n(r i)i=1,···,N.(8b) Now we introduce a new Lax type equation given byLτk =[B k+Ni=1q i∂−1r i,L].(9a)withq i,tn=B n(q i),r i,t n=−B∗n(r i)i=1,···,N.(9b) We have the following lemma[20].Lemma1.[B n,q∂−1r]−=B n(q)∂−1r−q∂−1B∗n(r).Proof.Without loss of generality,we consider a monomial:P=a∂n(n≥0).Then[P,q∂−1r]−=aq(n)∂−1r−(q∂−1ra∂n)−.Notice that the second term can be rewritten in the following way(q∂−1ra∂n)−=(q∂−1∂(ra)∂n−1−q∂−1(ra)′∂n−1)−=(−q∂−1(ra)′∂n−1)−=···=(−1)n q∂−1(ar)(n)=q∂−1P∗(r),then the lemma is proved.Proposition1.(3)and(9)give rise to the following new multi-component KP hierarchyB n,τk −(B k+Ni=1q i∂−1r i)t n+[B n,B k+N i=1q i∂−1r i]=0(10a)q i,tn=B n(q i),(10b)r i,tn=−B∗n(r i),i=1,···,N.(10c)Proof.We will show that under(9b),(3)and(9a)give rise to(10a).For convenience,we assume N=1,and denote q1and r1by q and r,respectively.By(3),(9)and Lemma1,we haveB n,τk =(L nτk)+=[B k+q∂−1r,L n]+=[B k+q∂−1r,L n+]++[B k+q∂−1r,L n−]+=[B k+q∂−1r,L n+]−[B k+q∂−1r,L n+]−+[B k,L n−]+=[B k+q∂−1r,B n]−[q∂−1r,B n]−+[B n,L k]+=[B k+q∂−1r,B n]+B n(q)∂−1r−q∂−1B∗n(r)+B k,t n=[B k+q∂−1r,B n]+(B k+q∂−1r)t n.Under(10b)and(10c),the Lax representation for(10a)is given byψτk =(B k+Ni=1q i∂−1r i)(ψ),(11a)ψtn=B n(ψ).(11b) Now,we list some examples in the new mcKP hierarchy(10).Example1(Thefirst type of KPSCS).For n=2and k=3,(10)yieldsu1,y−u′′1−2u′2=0,(12a)2u1,t−3(u2+u′1)t2+3u′′2+u′′′1−6u1u′1+2Ni=1(q i r i)′=0,(12b)q i,t2=q′′i+2u1q i,r i,t2=−r′′i−2u1r i.(12c)Set y:=t2,t:=τ3,u:=u1,and eliminate u2by differentiating the second equation with respect to x,we get thefirst type of KP equation with self-consistent sources(1).The Lax representation of(1a)isψy=(∂2+2u)(ψ),ψt=(∂3+3u∂+(32u x)+Ni=1q i∂−1r i)(ψ).Example2(The second type of KPSCS).For n=3and k=2,(10)yieldsu1,y+Ni=1(q i r i)′−u′′1−2u′2=0,(13a)3(u2+u′1)y−2u1,t−u′′′1+3Ni=1(q′i r i)′+6u1u′1−3u′′2=0,(13b)q i,t3=q i,xxx+3u1q i,x+(3u2+3u′1)q i,(13c)r i,t3=r i,xxx+3u1r i,x−3u2r i.(13d) Let y:=τ2,t=t3,u:=u1,and eliminate u2by integrating thefirst equation with respect to x, we get the second type of KPSCS(2).This equation was introduced in[10],and rediscovered by source generating method[14].Under(13c)and(13d),the Lax representation for(2a)isψy=(∂2+2u+Ni=1q i∂−1r i)(ψ),ψt=(∂3+3u∂+(32u x+3the equations yield12u1,xxxy+32u1,xyy−2(u21)xxx+12(qr)xxx=0,q t=(∂4+4u1∂2+(4u2+6u′1)∂+(4u3+6u′2+4u′′1))qr t=−(∂4+4u1∂2+(4u2+6u′1)∂+(4u3+6u′2+4u′′1))∗r.3ReductionsThe new mcKP hierarchy(10)admits reductions to several well-known(1+1)-dimensional systems.3.1The n-reduction of(10)The n-reduction is given byL n=B n or L n−=0,(14) then(6)implies thatB n(q i)=L n q i=ζn i q i,(15a)−B∗n(r i)=−L n∗r i=−ζn i r i.(15b)By using Lemma1and(15),we can see that the constraint(14)is invariant under theτk flow(L n−)τk =[B k,L n]−+Ni=1[q i∂−1r i,L n]−=[B k,L n−]−+Ni=1[q i∂−1r i,L n+]−+N i=1[q i∂−1r i,L n−]−=Ni=1[q i∂−1r i,B n]−=−N i=1(q i,t n∂−1r i+q i∂−1r i,t n)(16)=−Ni=1(ζn i q i∂−1r i−ζn i q i∂−1r i)=0.(17)The equations(14)and(5)imply thatφt n=0,so(L k)t n=0,which together with(17) means that one can drop t n dependency from(10)and obtainB n,τk=[(B n)k4u xxx+Ni=1(q i r i)x=0,q i,xx+2uq i=ζ2i q i,r i,xx+2ur i=ζ2i r i,i=1,···,N, with Lax representation(∂2+2u)(ψ)=λψ,ψt=(∂3+3u∂+3The first type of KdV equation with self-consistent sources can be solved by the inverse scattering method [12,21]and by the Darboux transformation (see [22]and the references therein).For n =3and k =2,(18)presents the first type of Boussinesq equation with self-consistent sources (t :=τ2,u :=u 1)u tt +12D −1u y +32N j =1q j r j )=ζ3i q i ,r i,xxx +3ur i,x −r i (32u x +3k+,B k+N i =1q i ∂−1r i,(19a)q i,t n =(B k +N j =1q j ∂−1r j )nk∗+(r i ),i =1,···,N,(19c)which is the so-called k -constrained KP hierarchy [18–20].For k=2and n=3,(19)gives rise to the second type of KdV equation with self-consistent sources.u t=14Ni=1(q i,xx r i−q i r i,xx),q i,t=q i,xxx+3uq i,x+32u x q i,r i,t=r i,xxx+3ur i,x−32u x r i,i=1,···,N.For k=3and n=2,(19)gives rise to the second type of Boussinesq equation with self-consistent sources.u tt+13Ni=1(q i r i)xx=0,q i,t=q i,xx+2uq i,r i,t=−r i,xx−2ur i,i=1,···,N.4ConclusionsA method is proposed in this paper to construct a new mcKP hierarchy,which enables us tofind thefirst and second types of KPSCS(i.e.,(1)and(2))in a different way from those in[10–14]and to get their Lax representations directly.The new mcKP hierarchy offers natural reductions to the well-known Gelfand-Dickey hierarchy with self-consistent sources and to the k-constrained KP hierarchy.The k-constrained KP hierarchy includes the second type of KdV equation with self-consistent sources(the Yajima-Oikawa equation)and the second type of Boussinesq equation with self-consistent sources.The method proposed here provides a general way to construct the soliton equations with self-consistent sources and their Lax representations.This approach for constructing multi-component hierarchies can be applied to other(2+1)-dimensional systems(such as mKP,BKP,CKP,etc)and semi-discrete systems(e.g.,2-Toda hierarchy).We will present some other new multi-component hierarchies in the forthcoming paper.AcknowledgmentsThis work was supported by National Basic Research Program of China(973Program) (2007CB814800)and National Natural Science Foundation of China(grand No.10601028).References[1]E.Date,M.Jimbo,M.Kashiwara,T.Miwa,J.Phys.Soc.Japan,50(11)(1981)3806.[2]M.Jimbo,T.Miwa,Publ.Res.Inst.Math.Sci.,19(3)(1983)943.[3]M.Sato,Y.Sato,In:Nonlinear partial differential equations in applied science(Tokyo,1982),pp259–271.North-Holland,Amsterdam,1983.[4]E.Date,M.Jimbo,M.Kashiwara,T.Miwa,Publ.Res.Inst.Math.Sci.,18(3)(1982)1077.[5]V.G.Kac,J.W.van de Leur,J.Math.Phys.,44(8)(2003)3245.[6]J.van de Leur,J.Math.Phys.,39(5)(1998)2833.[7]H.Aratyn,E.Nissimov,S.Pacheva,Phys.Lett.A,244(4)(1998)245.[8]R.Hirota,Y.Ohta,J.Phys.Soc.Japan,60(3)(1991)798.[9]S.Kakei,Phys.Lett.A,264(6)(2000)449.[10]V.K.Mel’nikov,Lett.Math.Phys.,7(2)(1983)129.[11]V.K.Mel’nikov,Comm.Math.Phys.,112(4)(1987)639.[12]V.K.Mel’nikov,Phys.Lett.A,128(9)(1988)488.[13]T.Xiao,Y.Zeng,J.Phys.A,37(28)(2004)7143.[14]H.-Y.Wang,Some Studies on Soliton Equations with Self-consistent Sources,PhDthesis,Academy of Mathematics and System Science,Chinese Academy of Sciences, 2007.[15]L.A.Dickey,Soliton equations and Hamiltonian systems,World Scientific PublishingCo.Inc.,River Edge,NJ,second edition,2003.[16]T.Xiao,Y.B.Zeng,Physica A,353(2005)38.[17]M.Antonowicz,Phys.Lett.A,165(1)(1992)47.[18]B.Konopelchenko,J.Sidorenko,W.Strampp,Phys.Lett.A,157(1)(1991)17.[19]Y.Cheng,J.Math.Phys.,33(11)(1992)3774.[20]L.A.Dickey,Lett.Math.Phys.,34(4)(1995)379.[21]R.L.Lin,Y.B.Zeng,W.-X.Ma,Physica A,291(2001)287.[22]R.L.Lin,H.S.Yao,Y.B.Zeng,Symmetry,Integrability and Geometry:Methods andApplications,2(2006)096.。
