Similarity Reduction Analysis for a Coupled KdV System

Commun. Theor. Phys. (Beijing, China) 48 (2007) pp. 399- 404@ International Academic PublishersVol. 48, No. 3, September 15, 2007Similarity Reduction Analysis for a Coupled KdV System*QIAN Su-Ping1,2,t and TIAN Li-Xin2'Department of Mathematics, Changshu Institute of Technology, Changshu 215500, China2Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang 212013, China(Received September 4, 2006)A bstract Using the direct method for a coupled KdV system, six types of the similarity reductions are obtained. Thegroup explanation of the results is also given. It is pointed out that, in order to find all the results by nonclassical Lieapproach, two additional condition equations should be satisfied at the same time together with two original equations.PACS numbers: 02.30.Jr, 02.30.Ik, 05.45.Yv, 47.32.-y, 47.35.+iKey words: coupled KdV equations, direct method, nonclassical Lie approach1 IntroductionIn this paper, we study a new integrable coupled KdVequation system:tion has been widely used in various natural science fields△{})=ut +ux-(k1- k2 + k(k2)uxespecially in almost all branches of physics. Some kindsof coupled KdV equations have also been introduced in+ k1VUx + k1W0x- VUx = 0,the literature such as one describing two resonantly inter-0?)=v+Uxxr +(k1- kz- 1)vvx + k20Uxacting normal modes of internal-gravity-wave motion ina shallow stratified liquid.1 In principle, many of other+ kzUUx - k1k2Ulx = 0,(2)coupled KdV equations are introduced mathematically be-where k1 and k2 are two arbitrary constants. It is clearcause of their integrability.l2,3] In Ref. [4], by using a long-that the coupled KdV system (2) is a special case for thewave approximation, Lou et al. derived some new typesphysical systems (1). Because of the important applica-of coupled KdV equation systems with some arbitrary pa-tions of the model and because the model has not yet beenrameters from a two-layer fluid model:deeply studied, especially, there is no study on its groupUt + Q1VUx + (a202 + Q3WV + Q4Uxx + o5u2)x= 0,invariant solutions, here, we use CK's direct method toVt +δ10Ux + (δ2u2 + δ3u0 + δ4Vx +δ502)x = 0,(1reduce the partial differential equation (PDE) system toan ordinary differential equation (ODE) system. A groupoceanic phenomena such as the atmospheric blockings,theoretical explanation of the results obtained by the di-the interactions between the atmosphere and ocean,5l therect method is provided in terms of nonclassical symmetryoceanic circulations and even hurricanes or typhoons.reduction method.The study of symmetries is one of the most powerfulmethods in every branch of natural science especially in2 Similarity Reductions from Direct Methodintegrable systems because of the existence of infinitelyThe CK's direct similarity reduction method impliesmany symmetries. To find the Lie point symmetry groupthat it is possible to find full symmetry groups of the Cou-of a nonlinear system, there is a standard method thankspled KdV system by a simple direct method without usingto the famous first fundamental theorem of Lie.7] And theany group theory. All the similarity solutions of the formstandard method had been widely used to find Lie pointsymmetry algebras and groups for almost all the known in-u(x,t) = U(x,t, P(ξ(x, t)(x,))),tegrable systems. However, to find non-Lie point symme-0(x,t)= V(x,t, P(x, t),(x,))，(3)try groups is still quite difficult. Furthermore, even for thewhere U, V, P, Q, and ξ are functions of the indicatedLie point symmetry groups, the final known expressionsvariables and P(E) and Q(ξ) satisfy an ordinary differen-may be quite complicated, and difficult for real applica-tial equation, may be obtained by substituting Eqs. (3tions especially for physicists and other non-mathematicalscientists. In Ref. [8], Clarkson and Kruskal (CK) intro-into Eqs. (2). However, similar to single partial differen-duced a simple direct method to derive symmetry reduc-tial equation case, one can prove that it is sufficient totions of a nonlinear system without using any group the-seek a similarity reduction of Eqs. (2) in a special form: .ory. For many types of nonlinear systems, the method canu(x,t)= ar(x,t) + Br(x, t)P(x,)),be used to find all the possible similarity reductions. Thisv(x,t) = a2(x,t) + B2(x, t)(x,t),(4)fact hints that there is a simple method to find generalizedsymmetry groups for many types of nonlinear equations.rather than the most general forms (3). .*The project supported by National Natural Science Foundation of China under Gr中国煤化工Science Foundationof Jiangsu Province under Grant No. BK2002003, the Project of Technology InnovatiYHC N M H GJiangsu Province inYear 2006 under Grant No. 72, and the Natural Science Directed Foundation of the Jiangsu Higher Education Institutions under GrantNo. 06KJD1 10001t E-mail: qsp@cslg.edu.cn.400QIAN Su-Ping and TIAN Li-XinVol. 48Substituting Eqs. (4) into the equation△| = 0 leads toβ$3P" + 3ξx(βrξx)xP" +[β1ξt + β1ξxxx + kqa2βrξx- (k1- k2 + ki kx2)a1β1ξx+ 3(β1xξx)x]P'-(k1- k2 + kikx)βξx PP' + k;μβ2β1ξaQP' + k1β1μβ2ξx PQ'-险ξxQQ' + (kra1-a2)B2ξxQ' - (k1- k2 + k1kz2)βrβBrxP2 + k1(β1$2)x PQ+ [3rt + Birrr -(ki - k:2 + kikz)(anB1)x + k1(a2β1)x]P - B2B2xQ2 - [(a2B2)x- k1(a12)x]Q+ [ant + Q1xxx- (ki - k2 + k1k2)a1Q1x + kr(a1Q2)x - a2Q2x]=0,(5)where the primes are derivatives with respect to ξ. Equation (5) is an ODEof {P,Q} if and only if the ratios of thecofficients of different derivatives and powers of P and Q are functions ofξ . That is to say, the following constrainedconditions for ξx≠0 must be satisfied:(B132)x = T6(ξ)Br$2,(11)β1β2ξx = T1(E)βζ3,(6)(a2B2)x - k:1(a132)x = F7(ξ)Brξ2,(12)β陉ξ∞= F2(ξ)Br5,(7(B1ξx)xξx = T8(ξ)β1ξ2，(13)(k1a1 - a2)B2ξx = T3()Brs2，(8β2B2x = F4(ξ)β1rζ3 ,(9β陉ξx = Fo(ξ)Brζ3 ,(14)ββ2ξ = Ts(E)βrζ2,(10)β1βrx = Tno()B15,(15)β1ξt + Brxxx + kra2Brξμ -(ki - k2 + k1k2)ar1βnξx + 3(3x2ξx)x = Fn(E)β1ζ ,(16) .βit + β1xx- (k1 - k2 + kh1k2)(a1β1)x + kiq(a2B)x = T12(E)B1ζ2，(17)a1t +a1xxx-(k1 - k2 + ki1k2)a1Q1x + ki1(a1a2)x - a2Q2x = F13(ξ)βrζ ,(18)where T;(ξ) (i= 1,2... 13) are some arbitrary functions of ξ to be determined. In the determinations of a1, β, a2,β2，P and Q, there exist some freedoms without loss of generality.Remark 1 If a1(x,t) has the form 01 = ao(x,t) +of Eqs. (6)~ (8):βr(x, t)0(E)，then one can take n(ξ) = 0 (by substitutingF3=T4=Ts=Tr=Tg=T1o=F11=0,P(E)→P(E)- ()).Remark 2 If a2(x,t) has the form a2 = ao(x,t) +T1=I2=Ts=Tg= 1,(19) .B2(x, t)0(E), then one can take几(ξ) = 0 (by substitut-ξ=0(t)x +σ(t)，βi=02(t)， B2=02(t)， (20)ing Q(E)→Q(S) - 2()).θ()x+o'(t)Remark 3 If Br(x,t) is given by B1 = Bo(x, t)8(E), thena1= (1-1)(2 - k)()’one can take Q(ξ) = const = So (by taking P(ξ) - -k:1(0'(t)x + o'(t))P(ξ)82o/S2(ξ)).02= (ki- 1)(k2- k1)0(t) '(21)Remark 4 If B2(x,t) is given by β2 = Bo(x, t)2(E), then0'(t)= T1204(t),(22)one can take 9(E) = const = So (by taking Q(E)→Q()20/()).0(t)0"(t) - 2(0"()])x) + 0(t)o"(t) - 20'(t)o'(t)Remark 5 If ξ(x, t) is determined by an equation of the(k1 - 1)(k2 - k1)02(t)form Sn(E) = ξo(x,t), then one can take 8(ξ) = ξ (by= T130(t),(23) .takingξ→5-'(E)).It is worth while to point out that each freedom willwhere T12= A is a constant.be fixed by using the corresponding remark once; in otherTo discuss further, two situations should be studiedwords, more uses of the remarks may result in loss ofseparately.generality. Similar to Refs. [8] and [9], by using ReSituation 1: A ≠0.marks 1 ~ 5 one can easily obtain the general solutionIn this case, equations (20) ~ (23) becomet(t-to)+t2_ (k1- 1)(k2- ki)B .ξ=[3A(t - to)]/3(t - to)1/3中国煤化工413)1201=-3(k1 - 1)(k2- k1)(t-to) 3(2/3)(k1 一1)IHCNM HGki1xh14(1/3)t2 - 2t(t一to)]02=-;3(k1 - 1)(k2- k1)(t-to) 3(2/3)(k1 - 1)(k2 - k1)(t - to).No.3Similarity Reduction Analysis for a Coupled KdV System401Br =[3A(t- to)2/3’32 =[3A(t-to)2/3’(24)where to, t1, t2, A, and B are arbitrary constants. Then the similarity solution of the coupled equation system (2takes the form:P()A(1/3)[t2- 2t1(t- to)] .[3A(t-to)]2/3 3(k1-1)(kz-k1)(t-to)f 3(2/3)(k1- 1)(k2- k1)(t-to)Q(ξ)kixk:A(1/3)[t2- 2t1(t - to)](25)“B3A(t-+0)23- 3(k1- 1)(k2-kr)(t-to)+ 3(2/3)(k1- 1)(k2- k1)(t-to)The similarity reduction equations are2A2ξP"= [(k1- k2+ kqk2)P- kiQ]P' +(Q-k1P)Q'-AP(k1- 1)(k2- k1)B,2k1A2ξQ" =[(k2+1- ki)Q2 - k2P)Q' +(Irk2P - kaQ)P' -4Q-(k1-1)(kz- h1). k;B(26)LetF(E) + 2AξG(ξ) + 2AξQ()=F(E)+ 2Aξk2(G() + 2AE)(ki1- 1)(k2-1)T (k2- 1)(k2-k1)’(k1- 1)(k2-1) (k2- 1)(k2- ki1)one can find thatF"=(F +2A{)F' + AF-(k1- 1)(kx-k)B，G"=(G + 2A{)G' + AG - (k1- 1)(k2- kx1)B,(27)Namely, F(E) and G(E) satisfy the same ODE. Integrating one of the equations in Eq. (27) once leads to a second-orderequation:6[(k1- 1)(k2- k1)B- AF]F" +3A(F')2 + 3|[2Aξ- (k1- 1)(k2- k1)B]F2+ 2AF3- 12AB(k1 - 1)(kz-ki)ξF +6(k1- 1)2(kx-k1)2B2ξ +C1=0.(28)Especially, let B = 0 in Eq. (28), equation (28) becomes1F"=!+ AξF(29)where C1 is an integration constant. This reduction equation is just the P34 (the 34-th equation in the Painleve/Gambierclassification) which is the Miura transformation of the Painleve II equation.l10]Situation2: A= 0In this case, equations (20) ~ (23) become:ξ=tox+ B(k1- 1)(kz- k1)t8t2/2+tt+t2， β=场,β2=t,tk1t1a1= Btit +(k1 - 1)(k2 - k:1)to'Q2= kiBtet +(hi - 1)(kz - k1)to '(30)where to, t1, t2, and B are arbitrary constants. The corresponding similarity solution of the coupled equation system(2) has the form:t1u= t场P()+ Btgt +.(ki1 - 1)(k2 - h1)to'v=场Q()+kiBtgt+(k1- 1)(k2 - k1)to .(31) .P"= [(kx1-k2+k1k2)P- kxQ]P'+(Q-kyP)Q'-B，Q"= [-(k1-k2-1)Q - k2PlQ' +(kr1k2P- k2Q)P'-kB. (32)It is a special case of Eqs. (26).All discussions above are valid only for ζx≠0. When ξx = 0, i.e., ξ = ξ(t), we can take ξ = t simply, thanks toRemark 5. For ξ = t, the following constrained conditions must be satisfied:β1β1x = F1(t)β1，(B1B2)x = F2(t)B1，(a2B2)x - ki(a1/$2)x = T3(t)β，β2B2x = T4(t)β,βlt + Prmr- (k1- k2 + kik2)(a1β)x + k1(a2β1)x = Fs(t)31, .Q1t +a1xrr-(k1- kr + k1k2)a1Q1x + kr(a1Q2)x - a2Q2x = T6(t)Br,βrβ1x= Fr(t)B2，(31B2)x = Ts(t)32，k1(a1B1)x - (a2B1)x = Fo(t)B2，β2βB2x = T1o(t)B2，B2t + Bxxx + (k1 - k2 - 1)(a2B2)x + k2(Q1B2)x = FHu(t)32 ,中国煤化工(33)CNM HGwhere T:(t) (i= 12...,12) are arbitrary functions of t to be determined. olunliar to une ζx于U Case, using Remarks1 ~ 5, one can easily obtain other two possible nonequivalent solutions of Eqs. (33). Here we only list the resultsinstead of derivations..402QIAN Su-Ping and TIAN Li-XinVol. 48Situation 3: The third reduction solution has the formu=(x+tut +o)P()+ (-1)(e2-k)'tiV=(x+tt + to)P()+kit1(34)(k1- 1)(kz- k1)where P(t) and Q(t) satisfy the reduction ODEs:P'=(k1-k2 + kink2)P2-k1PQ+}Q2，Q' =-(ki-k2- 1)Q2 - 2k2PQ + 2k1k2P2(35)with general solution2k2P=--. (36)(kz- 1)(k2-k1)t-t2- (k1-1)(kz-k1)t-ts'(kz-1)(kz-k1)t-t2~ (k1-1)(k2-1)t-t3'Situation 4: The last reduction solution readst1k1x(37)(kr- 1)(k2- k1)(t-to) + (t-to)’"=(k1- 1)(k2-k)(t-to)+(t-to) .where to, t1, t2, and t3 are arbitrary constants.3 Reductions Based on Conditional Symmetry MethodIn various single PDE cases, all the similarity reductions can also be obtained by the“nonclassical method” ofBluman and Cole.11] In this paper we would like to apply the nonclassical method to the coupled KdV system (2).According to the nonclassical method, we shall now look for a transformation group leaving simultaneous solutions offour equations invariant, namely Eqs. (2), and0g}=Tu + Xux-U,△{= Tvr+ Xvx-V,(38)where T, X, U, and V are the same as those that appear in the following infinitesimal transformationsx=x+εX(x,t,u,v)，t=t+ eT(x,t,u,v)，u= u+eU(x,t,u,v)，v=v+εV(x,t, u,v),(39)with ε being an infinitesimal parameter. A general vector field on R2 x R2 takes the formTT 0u=Xx+Tx+Unu+V，(40)and need to know the first prolongation pr(1)y of y, third prolongation pr(3)y. The infinitesimal criterion of invarianceof Eqs. (2) and (38) under the group (39) is given byprl1)△{?)|(i,j=3,4, k= 1,2)，(41)0})=0,0(3)=0pr/3)△{3)|ls}"=0,O(3)=0=0(i,k=1,2, j=3,4).(42)Equations (41) are satisfied identically. The remaining equations lead to a set of determining equations that must thenbe solved. For further discussions, we consider some diferent subcases.Case 1 T≠0. In this case, one can take T = 1 simply without loss of generality. After substituting allthe extensions into Eqs. (42), compare the coefficients of linearly independent expressions of the type uux andvJvh (where uUt, Ut, Uxxx, and Vxxx have been cancelled by Eqs. (2) and (38)). The various constraint conditionsare obtained. We do not write these determining equations down because of its complicity. The solutions of thesedetermining equations have two subcases: .Case 1 (a)x- 3c1(k1 - 1)(k2- k1)t + 3c2U=- -2u + 3c1、. V=--2u + 3k1C1(43) .3(t - co)3(t-co)3(t- Co)with Co, C1, C2 being arbitrary constants, i.e.,_x- 3c1(k1- 1)(k2-ki)t +3c2.8.8 ， 3c1-2u 8.3k1C1-2v 8(44)3(t- co)ox T OtT 3(t-co)OuT 3(t-co) 8Then, the global general invariant can easily be obtained. This subcase is just the result of classical Lie approach. Thecorresponding similarity reduction coincides with Situation 1 of Sec. 2 for2A1/3t1 .co= to，C2=-3(2to + t2) .32/3(k1- 1)(k2- k1)’Case1(b)中国煤化工X=-c(k1- 1)(kx2-k1)t+C2，U=c1,MHCNMH G(45)i.e.r=[-c(ki-1)(k2 -k)t+c2lx+o +1n.C1h18(46).No.3Similarity Reduction Analysis for a Coupled KdV System403where C1 and C2 are arbitrary constants. This subcase can also be obtained from the classical Lie approach and thecorresponding similarity reduction coincides with Situation 2 of Sec. 2 for C1 = tB, C2= -t1/to.Case 2 T= 0. In this case, without loss of generality, one can put X = 1. Similar to Case 1, substituting all theextensions into Eqs. (42), but the complexity of this equation is the reason why we cannot solve it. Thus we proceed,by making ansatz on the form of functions U(x,t,u,0) and V(x,t, u, 0), two different subcases are obtained (in thiscase, Ci (i = 0,1,2,3, 4) are arbitrary constants):Case 2(a)_c1X=1，T=0，U=x+ct+Co(k1- 1)(k2- k1)(x +crt + co)_k1c1(47)x+ct+Co (ki- 1)(k2- k1)(x+cit + co)i.e.88x”Lx+ct+Co (hi1- 1)(k2- k1)(x+ cnt + co)J Ouk1c11(48)(k1- 1)(k2- k1)(x+c1t + co)J 8vIn this subcase, one can get the similarity reduction of Situation 3 of the Sec. 2 for Co= to,c1= t1.Case 2(b)k(49)(ki- 1)(k2-k1)t-c1(k1- 1)(k2- kr)t-c1)0(50)):x (k1- 1)(k2- ki)t-c1 8u(k1-1)(kz-k1)t-c1) OuThis subcase can also be obtained by the classical Lie approach. Situation 4 of Sec. 2 is a special case of the corre-sponding similarity reduction.Case 2(c)X=1，T=0，x+k1Ct +k1C2 (k1 - 1)(kz- k1)(x+ kncit +c2)krukic1(51)x+ kicnt + k1C2(k1- 1)(k2- k1)(x + kic1t + C2)k1C11日)x t lx+k1c1t+h1C2(k1- 1)(k2- k1)(x+ k1c1t + c2)J8k1uK硫c118(52)Lx +kict+h1C2 (k1- 1)(k2z-k1)(x+ k1ct+c2)Jou’The related reduction solution of the coupled equation system (2) takes the form:u= (x+ kicnt + k1cx)U(t) +(ki- 1)(ka- hr)v= k1xU(t) + V(t), .(53)where U(t) and V(t) are solutions of the reduction ODEsU'=(k1- 1)(k2-k1)U2， V'=(k1- 1)(kz- kr)UV,(54)which can be solved exactly. The final solution of the coupled KdV system (2) in this subcase reads .x+ k1C2k1C1C3“，= (k1- 1)(k2-k1)t-C3- (k1- 1)(k2- kr)(x1- 1)(k2 - k1)t-c3] 'k1x- C4 _(55)(h1-1)(kz-k1)t-C3Case 2(d)k1x+ct + C2(k1- 1)(k2-k1)(kux士c1t +c2) 'k1v中国煤化工(56)k1x+cnt+C2 (k1- 1)(k2 - ki)(k1x +c1t +cHCNMH G_C18x TLkrx +ct+C2 (ki - 1)(k2- ki)(krx +cit + c2)J Fu ..40QIAN Su-Ping and TIAN Li-XinVol. 48k1C1(57) .lk1x +c1t+C2 (k1- 1)(k2- k1)(k1x+ct + c2)J 8vThen the solution of the coupled equation system (2) takes the formC1u=U(t)+xV()，v= (k1x +ct + c)V() +(k1-1)(k2-kn)’(58)where U(t) and V(t) satisfy the ODEs:U'=(k1- 1)(k2- k;)UV, V'= (k1- 1)(k2- k1)v2.(59)Then the coupled KdV system (2) has the exact solution:C4(k1- 1)(kz-kr)t-C3 T (k1- 1)(kz-k1)t-C3k1x +cnt+ c2(60)(k1- 1)(k2-k1)t-C3 T (k1- 1)(k2-k1)t-C3Case 2(e)X=1，T=0，U=-cq(kju-v)， V=-k1c(kru-v),(61)i.c0c1(k1u-v)二- kicl(kqu- v)(62))x)u)v’Then the solution of the coupled equation system (2) takes the form:_U(t) 」(k1C1x- 1)V(t)，v=U(t) + xV(t),(63) .1Ric1where U(t) and V(t) are satisfied by the ODEs:U'=(k1- 1)(k2 -k)uv，v'= (k1- 1)(k2 -k1)v2.(64)kThe coupled KdV system (2) then has the exact solution:C1C3+ 1(ki- 1)(k2- k1)t-kiC2T k1c([(k1- 1)(k2- k1)t- k1C2]k1xC3(65)"(k1- 1)(k2-k)t-kiC2十(k1- 1)(k2- k)t- kiC2The final three subcases, Cases 2(c) ~ 2(e), can only be obtained via nonclassical Lie symmetry group approach.4 SummaryIn this paper, by using a direct method, we have obtained four types of the similarity reductions of a new integrablecoupled KdV system. The reduction solutions obtained by the classical Lie approach are only the special cases:Situations 1, 2, and 4 obtained by the direct method. Situation 3 cannot be obtained by the classical Lie approach.However, after introducing two additional condition equations, we can also find all the results by using the standardalgorithm that provides the symmetry algebra, i.e., the Lie algebra of the Lie group of the local point transformationsleaving the joint solution sets of Eqs. (2) and additional conditions (39) invariant.AcknowledgmentsOne of the authors (S.P. Qian) is indebt to Prof. S.Y. Lou for his helpful discussions.References[6] S.Y. Lou, X.Y. Tang, M. Jia, and F. Huang, preprint[1] JA. Gear and R. Grimshaw, Stud. Appl. Math. 70 (1984)nlin.PS/0509039.235; J.A. Gear, Stud. Appl. Math. 72 (1985) 95[7] P.J. Olver, Application of Lie Group to Differential Equa-tion, Springer, New York (1986).[2] M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlin-[8] P.A. Clarkson and M.D. Kruskal, J. 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