An Extension of Algorithm on Symbolic Computations of Conserved Densities for High-Dimensional Nonlinear Systems

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1 Commn. Theor. Phys. (Bejng, Chna 50 (2008 pp c Chnese Physcal Socety Vol. 50, No. 1, Jly 15, 2008 An Extenson of Algorthm on Symbolc Comptatons of Conserved Denstes for Hgh-Dmensonal Nonlnear Systems YANG X-Dong, 1 RUAN Hang-Y, 1 and LOU Sen-Ye 1,2 1 Department of Physcs, Nngbo Unversty, Nngbo , Chna 2 Department of Physcs, Shangha Jao Tong Unversty, Shangha , Chna (Receved September 24, 2007 Abstract An mproved algorthm for symbolc comptatons of polynomal-type conservaton laws (PCLaws of a general polynomal nonlnear system s presented. The algorthm s mplemented n Maple and can be sccessflly sed for hgh-dmensonal models. Frthermore, the algorthm dscards the restrcton to evolton eqatons. The program can also be sed to determne the preferences for a gven parameterzed nonlnear systems. The code s tested on several known nonlnear eqatons from the solton theory. PACS nmbers: Jr, j, Wz Key words: conservaton laws, nonlnear systems, compter algebra 1 Introdcton The Koreteweg-de Vres (KdV eqaton, one of the most classcal and completely ntegrable model from the solton theory, possesses many remarkable mathematcal propertes sch as nfntely many conservaton laws, [1] nfntely many symmetres, [2] Bäcklnd transformatons, Darbox transformatons, b-hamltonan strctres, [3,4] nverse scatterng transformatons, [5,6] and etc. Among these notable propertes, the exstence of nfntely many ndependent conservaton laws (CLaws s an mportant lnk to the dscovery of the other specal propertes. Frthermore, t predcts ntegrablty. Natrally, the work of constrctng the nfntely many conservaton laws plays an mportant role n the solton theory. A very extensve stdy of CLaws s fond n Ref. [7], whch ncldes both Lagrangan and Hamltonan formlatons. However, comptng CLaws wth paper and pensl s a challengng task becase t nvolves tedos calclatons. Recently, there are varos methods to compte CLaws of nonlnear systems. [8] By sng scalng propertes, Ito et al. developed a REDUCE program CONSD [9] to compte PCLaws and then the mproved verson SYMCD. [10] In the same way, Goktas and Hereman also developed a package condens.m [11] n Mathematca to search for PCLaws for nonlnear polynomal partal dfferental eqatons (PDEs. Bocharov and coworkers desgned a package DELA [12] n Trbo PASCAL wth Le approach. Sanders et al. represented a software package n Maple and FORM. [13,14] They sed an extenson of total dervatve operator to a Hesenberg algebra, whch allowed them to nvert the total dervatves on ts mage. Thomas Wolf developed a REDUCE package CONLAW1-4. [15] Hs approach s based on solvng over-determned systems of PDEs, whch shows p n the comptaton of Le-pont symmetres and generalzed symmetres. Yao et al. mproved the algorthm gven by Goktas and Hereman and desgned a maple package CONSLAW, [16] whle Hereman extended the algorthm nto mlt-dmensons by sng Eler Operator and Homotopy Operators. However, the algorthm gven by Hereman s only for the evolton systems. [17] To or knowledge, there exst many hghdmensonal ntegrable models from the solton theory that are not evolton systems or cannot be converted nto evolton cases. However, t s defntely tre that some of them possess PCLaws. In ths paper, we ntrodce an algorthm that s based n part on deas presented n Hereman [11,17] and Yao. [16] Or algorthm can be sed to comptng the PCLaws of general polynomal nonlnear systems. The algorthm has the advantage that s fary straghtforward to mplement n any symbolc langage. We also present a software package n Maple, whch atomates the tedos comptaton for the constrcton of conserved denstes. The paper s organzed as follows. In Sec. 2, we otlne the algorthm and sbrotnes. In Sec. 3, several examples are gven to demonstrate the applcatons. Then, a short smmary s gven n Sec Comptaton of Conserved Denstes 2.1 Defnton For smplcty, consder a sngle (n + 1-dmensonal nonlnear eqaton, H( N x 1 1 xn 2 2 xn n n x N = 0, (1 where = (x 1, x 2, x 3,...,x n, x s a potental fncton wth n-dmensonal spatal varables x 1, x 2,...,x n and one-dmensonal tme varable x, and H s a polynomal of (N 1+N 2 + +N x N 1 1 xn 2 2 xn n n x N x N 1 1 xn 2 2 xn nder dfferent {N 1, N 2,...,N }. A conservaton law s of the form ρ x + n (J x = 0, (2 =1 whch s satsfed for all soltons of Eq. (1. The fnctonal ρ s the conserved densty, J s the assocated flx The project spported by the Scentfc Fnd of Edcaton Department of Zhejang Provnce of Chna nder Grant No , the Natonal Natral Scence Fondatons of Chna nder Grant Nos , , and , the State Basc Research Program of Chna (973 Program nder Grant No. 2007CB814800, and the K.C. Wong Magna Fnd n Nngbo Unversty

2 24 YANG X-Dong, RUAN Hang-Y, and LOU Sen-Ye Vol. 50 wth respect to the spatal varable x. As we all know, the notable mkdv eqaton t + 2 x + x 3 = 0, (3 possesses nfntely many conservaton laws and the frst three densty-flx pars are ( t + ( 3 /3 + x 2 x = 0, ( 2 t + ( 4 /2 2 x + 2 x 2 x = 0, ( x t + ( x x x 2 12 x x 3 x = 0. (4 It s clear that the conserved denstes ρ =, 2, and x are all nvarant nder the scalng symmetry transformaton (x, t, (λ 1 x, λ 3 t, λ 1, (5 where λ s a parameter. The scalng nvarance, whch s a specal Le-pont symmetry, s an ntrnsc property of many ntegrable nonlnear PDEs. Accordng to Eq. (5, wthot loss of the generalty, one can ntrodce the rank of varables (ncldng the potental fncton and dependent varables x and t as follows: Ω( = 1, Ω(x = 1, Ω(t = 3. (6 Two clear and mportant reslts shold be mentoned here. The frst s all monomals n Eq. (3 are nform n rank 4, whle the second s the frst three conserved denstes and the correspondng flxes are nform n rank 1, 2, 4 and 3, 4, 6 respectvely, whch satsfy Ω(J = Ω(ρ + Ω(x Ω(t, (7 where Ω(ρ and Ω(J are the rank of conserved densty and the assocated flx respectvely. Unform n rank plays an mportant role n or algorthm. Or algorthm explots ths dea n the constrcton of conserved denstes. 2.2 Algorthm The compter algebrac algorthm and rotnes to constrct the CLaws for Eq. (1 are descrbed as the followng steps Determne Weghts of Varables and Parameters The work of determnng the weghts of varables and parameters can be easly done by the defnton of nform n rank. In or algorthm, we defne the weght of partal dervatves ( as follows: N 1+N 2 + +N Ω x N 1 1 xn 2 2 xn = Ω ( =1 xn = Ω( [N Ω(x ], (8 =1 where denotes the arbtrary potental fncton and x s one of ts ndependent varables. For example, the ranks of the three terms n Eq. (3 are Ω( Ω(t, 3Ω( Ω(x, Ω( 3Ω(x, respectvely. The reqrement of nformty n rank leads to Ω( Ω(t = 3Ω( Ω(x = Ω( 3Ω(x, whch yelds Ω( = Ω(x, Ω(t = 3Ω(x, and Ω(x can be an arbtrary constant. In or algorthm, let Ω(x = 1, one can drectly obtan an general rank solton (6. However, there ndeed exsts sch a case that the eqaton s not nform n rank. Fortnately, we can se a trck gven by Hereman that ntrodces an axlary parameter wth (nknown weght. [11] Here, we constrct a sbrotne W eght(eqset :: set, P araset :: set, where EqSet denotes a set of eqatons and ParaSet s a set of parameters wth (nknown weght. Takng the ( dmensonal KD eqaton, t = x 3 + 6β x 3 2 α2 2 x + 3w y + 3α x w, w x = y, (9 as an example, where the parameter β has nknown weght, one shold call the sbrotne as Weght({ t = x 3 + 6β x 3 2 α2 2 x + 3w y + 3α x w, w x = y }, {β}. In detal, the procedre proceeds as follows: ( Collectng varables of EqSet and parameters n ParaSet, we derve the set S0 = {, w, x, y, t, β}. ( Gatherng all the monomals of each of eqatons from the system, we derve the set S1 = {{ t, x 3, 6β x, 3 2 α2 2 x, 3w y, 3α x w }, {w x, y } }. ( Selectng all the partal dervatves of the eqaton, we form the set S2 = { t, x, x 3, y, w x, w y }. (v Replacng each element n S2 wth formla N 1+N 2 + +N x N 1 1 xn 2 2 N 1+N 2 + +N xn x N 1 1 xn 2 2 xn =, =1 xn then we have { the set S3 = t = t, x = x, x 3 = x 3, y = y, w x = w x, w y = w }. y (v Sbstttng S3 nto S1, we derve a new set {{ S4 = t, x 3, 6β2 x, 3α2 3 2x, 3w y, 3αw x { w x, y }}. (v Replacng each element n S4 wth formla f ln(f, we have the set S5 = {{ ln( ln(t, ln( 3 ln(x + πi, ln(β + 2 ln( ln(x + ln(6 + πi, 2 ln(α + 3 ln( ln(x + ln ( 3 2, ln(w ln(y + ln(3 + πi, }, ln(α + ln( + ln(w ln(x + ln(3 + πi }, {ln(w ln(x, ln( ln(y + πi} }. (v Removng all the monomals wthot weght for each element n S5, we have the reslt S6 = {{ln( ln(t, ln( 3 ln(x, ln(β + 2 ln( ln(x, 3 ln( ln(x, ln(w ln(y, ln( + ln(w ln(x}, {ln(w ln(x, ln( ln(y}}.

3 No. 1 An Extenson of Algorthm on Symbolc Comptatons of Conserved Denstes for Hgh-Dmensonal Nonlnear Systems 25 (v Replacng each element n S0 wth formla f f = exp(ω f. Here Ω f does not denote partal dervatve bt an element of array Ω. Then we have the set S7 = { = exp(ω, w = exp(ω w, x = exp(ω x, y = exp(ω y, t = exp(ω t, β = exp(ω β }. (x Sbstttng S7 nto S6, we derve a new set S8 = {{Ω Ω t, Ω 3Ω x, Ω β + 2Ω Ω x, 3Ω Ω x, Ω w Ω y, Ω + Ω w Ω x }, {Ω w Ω x, Ω Ω y }}. (x Constrctng the rank eqaton wth the condton of nform n rank and addng the general assmpton Ω x = 1, yelds S9 = {Ω Ω t = Ω 3Ω x, Ω Ω t = Ω β + 2Ω Ω x, Ω Ω t = 3Ω Ω x, Ω Ω t = Ω w Ω y, Ω Ω t = Ω + Ω w Ω x, Ω w Ω x = Ω Ω y, Ω x = 1}. (x Solvng eqaton S9, we get the solton Sol = {Ω = 1, Ω w = 2, Ω x = 1, Ω y = 2, Ω t = 3, Ω β = 1}. (x Accordng to Sol, for cases shold be consdered. (a Sol s NULL, namely the eqaton s not nform n rank, then prompt the ser wth nformaton and fnsh the process. (b Sol has free weght, prompt the ser to enter a vale, then go on. (c Sol has fractal weght, then mltply each weght n Sol wth the least common mltple of the denomnators n the fractal weghts, then go on. (d otherwse, go on. (x Select fnctons and parameters wth postve weght from Sol, we have the set S11 = {Ω = 1, Ω w = 2, Ω β = 1}. (xv Selectng ndependent varables wth negatve weght from Sol and transform weght to postve, we derve the set S12 = {Ω x = 1, Ω y = 2, Ω t = 3}. (xv Unon set S11 and S12, then convert nto the form as Rset = {[, 1], [w, 2], [β, 1],[x, 1], [y, 2], [t, 3]}. (xv Convert each element n set S12 nto a lst as the form of [M, Rank(t-Rank(M], where M s a varable and Rank(M s the correspondng weght. Then collect the obtaned lst nto a set V set = {[x, 2],[y, 1], [t, 0]}. We pont ot that, Rset and V set are the fnal reslts whch wll be sed as the key parameters n the next sbrotne n constrctng the general form of conserved denstes Constrct the General Form of Conserved Denstes and Flx It s clear that all the monomals n P (T N x 1 ( 1 xn 2 2 xn n n x N W m x N 1 1 xn 2 2 xn n n x N are nform n rank [m Ω(W ] + [N Ω(x ], =1 (10 where W s a potental fncton or a weghted parameter, and Ω(X denotes the weght of the assocated varable X. In the above sbsecton, the weghts of varables and parameters are obtaned as the set Rset. Now, we tlze Rset to constrct the general form of conserved denstes. For a gven rank Rank, the procedre GenT erms(rset :: set, Rank :: posnt, Remove :: boolean wll atomatcally fnd all the possble combnatons nto a set as the form of {[m, N]}, whch satsfes [m Ω(W ] + [N Ω(x ] = Rank. =1 For smplcty, we take the (2 + 1-dmensonal breakng solton eqaton (BSE, xt = 4 x xy + 2 x 2 y x 3 y, (11 as another example. By sng the above algorthm, the program reports that BSE s a system wth free weghts. If we take Rset = {[, 1], [x, 1],[y, 2], [t, 4]} and Rank = 5 as an example, n detal, the procedre proceeds as follows: Step 1 Constrct all the possble combnatons nto the set {[m, N]}. ( Sort the Rset n descendng order n rank then convert t nto two lsts as the form L1 = [V 1, V 2,...,V n ] and L2 = [Ω(V 1, Ω(V 2,...,Ω(V n ]. We have two lsts L1 = [t, y,, x] and L2 = [4, 2, 1, 1]; ( For each element n L1, f V s a potental fncton or a weghted parameter then append ts address n L1 nto lst L3; f V s a tme varable then nsert t nto lst L4; otherwse (namely a spatal varable nsert nto L5. As a reslt, we derve L3 = [3], L4 = [1] and L5 = [2, 4]; ( Set S1 = {[Rank, [ ]]}; (v for from 1 to nops(l2 do Set S2 = {} Set vrank=op(, L2

4 26 YANG X-Dong, RUAN Hang-Y, and LOU Sen-Ye Vol. 50 for jj from 1 to nops(s1 do Set L6=op(jj, S1 Set rrank=op(1, L6 Set L7=op(2, L6 for kk from 0 to rrank/vrank do Set S2=S2 non {[rrank kk vrank, [op(l7,kk]} od od Set S1=S2 od Remarks Maple fncton nops(x: retrns the nmber of elements of a lst or set X. Maple fncton op(x: retrns all elements of a lst or set X. Maple fncton op(n, X: retrns the n-th element of a lst or set X. Now we have the set S1 = {[5, [0, 0, 0, 0]], [4, [0, 0, 0, 1]], [3, [0, 0, 0, 2]], [2, [0, 0, 0, 3]], [1, [0, 0, 0, 4]], [0, [0, 0, 0, 5]], [4, [0, 0, 1, 0]], [3, [0, 0, 1, 1]], [2, [0, 0, 1, 2]], [1, [0, 0, 1, 3]], [0, [0, 0, 1, 4]], [3, [0, 0, 2, 0]], [2, [0, 0, 2, 1]], [1, [0, 0, 2, 2]], [0, [0, 0, 2, 3]], [2, [0, 0, 3, 0]], [1, [0, 0, 3, 1]], [0, [0, 0, 3, 2]], [1, [0, 0, 4, 0]], [0, [0, 0, 4, 1]], [0, [0, 0, 5, 0]], [3, [0, 1, 0, 0]], [2, [0, 1, 0, 1]], [1, [0, 1, 0, 2]], [0, [0, 1, 0, 3]], [2, [0, 1, 1, 0]], [1, [0, 1, 1, 1]], [0, [0, 1, 1, 2]], [1, [0, 1, 2, 0]], [0, [0, 1, 2, 1]], [0, [0, 1, 3, 0]], [1, [0, 2, 0, 0]], [0, [0, 2, 0, 1]], [0, [0, 2, 1, 0]], [1, [1, 0, 0, 0]], [0, [1, 0, 0, 1]], [0, [1, 0, 1, 0]]} (v for each element [rrank, rlst] n set S1, f rrank = 0 then nsert rlst nto S3, omt t. From now on, we get all possble combnatons {[m, N]} as the reslt S3 = {[0, 0, 0, 5], [0, 0, 1, 4], [0, 0, 2, 3], [0, 0, 3, 2], [0, 0, 4, 1], [0, 0, 5, 0], [0, 1, 0, 3], [0, 1, 1, 2], [0, 1, 2, 1], [0, 1, 3, 0], [0, 2, 0, 1], [0, 2, 1, 0], [1, 0, 0, 1], [1, 0, 1, 0]}. Indeed, f the gven eqaton s an evolton one, the dervatve of potental fncton wth respect to t wll never appear ether n densty ρ or flx J. So one can omt the tem [t, Rank(t] n Rset to speed p. In constrctng the general form of densty ρ, some redndant terms whch can be wrtten as a total dervatve to a spatal varable x shold be removed as the reslt belong to the same eqvalence class. However, for flx J the work of removng terms s not necessary. One shold call the GenTerms wth Remove = tre for densty ρ or Remove = false for flx J. Step 2 Remove redndant terms whch can be wrtten as a total dervatve. Takng accont of formla (10 and sng the reslts L1, L3, L4, L5 and S3 whch are obtaned n Step 1, we have ( 0 x5 y 0 t 0 = 0, (1 x4 y 0 t 0 = x 4, (2 x3 y 0 t 0 = 6 x x x 3, ( 3 x2 y 0 t 0 = 62 x + 32 x 2, ( 4 x1 y 0 t 0 = 43 x, ( 5 x0 y 0 t 0 = 5, ( 0 x3 y 1 t 0 = 0, (1 x2 y 1 t 0 = x 2 y, ( 2 x1 y 1 t 0 = 2 x y + 2 xy, ( 3 x0 y 1 t 0 = 32 y, ( 0 x1 y 2 t 0 = 0, (1 x0 y 2 t 0 = y 2, (0 x1 y 0 t 1 = 0, (1 x0 y 0 t 1 = t, (12 and the constant term shold be omtted. Gatherng all the monomals on the rght-hand sde of eqaton (12, we derve the set rs1 = { x 4, x x 2, x 3, 2 x, 2 x 2, 3 x, 5, x2 y, x y, xy, 2 y, y 2, t }. Natrally, the canddate of the conserved flx of rank 5 can be wrtten as J = C J1 x 4 + C J2 x x 2 + C J3 x 3 + C J4 2 x + C J5 2 x 2 + C J6 3 x + C J7 5 + C J8 x 2 y + C J9 x y + C J10 xy + C J11 2 y + C J12 y 2 + C J13 t. Now we transform the terms (except for all N 0, = 1,...,n n Eq. (12 nto the form ( 1 x 4 = (( 1 x 3 x = ( x 3 x, ( 2 x 3 = (( 2 x 2 x = (2 x x x = (2 x 2 x + (2 2 x x, ( 3 x 2 = (( 3 x 1 x = (3 2 x x, ( 4 x 1 = (( 4 x 0 x = ( 4 x, ( 1 x 2 y 1 = ((1 x 1 y 1 x = ( xy x, ( 1 x 2 y 1 = ((1 x 2 y 0 y = ( x 2 y, ( 2 x 1 y 1 = ((2 x 0 y 1 x = (2 y x, ( 2 x 1 y 1 = ((2 x 1 y 0 y = (2 x y ( 3 y 1 = (( 3 y 0 y = ( 3 y, ( 1 y 2 = (( 1 y 1 y = ( y y. (13 Then we expand the rght-hand sde of Eq. (13 as ( x 3 x = { x 4}, (2 x 2 x = {2 x x x 3}, (2 2 x x = {4 x x 2}, (3 2 x x = {6 2 x x 2}, ( 4 x = {4 3 x }, ( xy x = { x 2 y}, ( x 2 y = { x 2 y}, (2 y x = {2 x y + 2 xy }, (2 x y = {2 x y + 2 xy }, ( 3 y = {3 2 y }, ( y y = { y 2}. (14 It s easy to fnd ot that the monomals grop comng from each { } n Eq. (14 can be constrcted as a total dervatve by a proper coeffcent combnaton, namely each term can be replaced by the other terms n the same

5 No. 1 An Extenson of Algorthm on Symbolc Comptatons of Conserved Denstes for Hgh-Dmensonal Nonlnear Systems 27 grop (namely a same eqvalence class. To be pont ot, all the eqvalence classes whch can be sed to remove the redndant terms have been constrcted as Eq. (14. Now, the only thng left for s to do s how to remove a proper term n each grop. For (1 + 1-dmensonal evolton eqaton(s, one of the most effcent way s to remove the term wth the hghest dervatve factor n each grop. However, ths dea wll never take effect n the system wth two or more varables. Then we have another way as follows: Transform rs1 nto two sets as the form ss1 = { x 4 = C 1, x x 2 = C 2, x 3 = C 3, 2 x = C 4, 2 x 2 = C 5, 3 x = C 6, 5 = C 7, x2 y = C 8, x y = C 9, xy = C 10, 2 y = C 11, y 2 = C 12, t = C 13 } and ss2 = {C 1 = x 4, C 2 = x x 2, C 3 = x 3, C 4 = 2 x, C 5 = 2 x 2, C 6 = 3 x, C 7 = 5, C 8 = x 2 y, C 9 = x y, C 10 = xy, C 11 = 2 y, C 12 = y 2, C 13 = t }. Sbstttng ss1 nto the rght-hand sde of Eq. (14, then we have a set ss3 = {C 1, 2C 2 + 2C 3, 4C 2, 6C 4 + 3C 5, 4C 6, C 8, 2C 9 + 2C 10, 3C 11, C 12 }. Solvng eqaton ss3, we get the solton ss4 = {C 1 = 0, C 2 = 0, C 3 = 0, C 4 = C 4, C 5 = 2C 4, C 6 = 0, C 8 = 0, C 9 = C 9, C 10 = C 9, C 11 = 0, C 12 = 0}. For each element n ss4, f ts left-hand sde s eqal to ts rght-hand sde, then nsert the left-hand sde nto ss5; otherwse nto ss6. Then we have the reslt ss5 = {C 4, C 9 } and ss6 = {C 1, C 2, C 3, C 5, C 6, C 8, C 10, C 11, C 12 }. It s clear that each term n ss6 can be expressed by ss5 wth dfferent combnaton of coeffcents, namely the correspondng terms wth respect to ss6 shold be removed from the reslt. Then sbstttng ss2 nto ss6 yelds rs2 = { x 4, x x 2, x 3, 2 x 2, 3 x, x 2 y, xy, 2 y, y 2}. The fnal reslt can be wrtten as rs3 = rs1, mns, rs2 = { 2 x, 5, x y, t }, and the canddate of conserved densty s ρ = C ρ1 2 x + C ρ2 5 + C ρ3 x y + C ρ4 t. The advantage of or algorthm s drect n comptaton and sccess n avodng the ntegral operaton, whch s a weakness of the most symbolc comptaton systems. The algorthm s descrbed as follows: cont:=0; for from 1 to nops(s3 do L8:=op(, S3; T:=1; for jj from 1 to nops(l3 do Compte T = W m ll:=op(jj, L3; T:=T op(ll, L1ˆop(ll, L8; f T <> 1 then f nops(l4>0 then Compte the delvatve of T respect to t ll:=op(1, L4; var:=op(ll, L1; dervatve varable nn:=op(ll, L8; f nn = 0 then T:=T; T:=dff(T, var$nn; dervatve order for jj from 1 to nops(l5 do Compte the delvatve of T respect to x ll:=op(jj, L5; var:=op(ll, L1; dervatve varable nn:=op(ll, L8; dervatve order f nn = 0 then P:=T; P:=dff(T, var$nn; exprlst0[]:=getterms(dsbs(eqset, P; f Remove then Compte the eqvalent class for jj from 1 to nops(l5 do expr:=t; ext:=0;

6 28 YANG X-Dong, RUAN Hang-Y, and LOU Sen-Ye Vol. 50 for kk from 1 to nops(l5 do ll:=op(jj, L5; var:=op(ll, L1; nn:=op(ll, L8; f jj=kk then f nn > 0 then nn:=nn-1; var0:=var; ext:=1; f nn<>0 then expr:=dff(expr, var$nn; f ext=1 then next; termlst:=getterms(expr; for kk from 1 to nops(termlst do term:=op(kk, termlst; cont:=cont+1; exprlst1[cont]:=dsbs(eqset, dff(term, var1; exprlst0[]:=[ ]; rs1:={seq(op(exprlst0, ll=1.. nops(s3}; f Remove then ss1:=seq(op(ll, rs1=c[ll], ll=1.. nops(rs1; ss2:=seq(c[ll]=op(ll, rs1, ll=1.. nops(rs1; ss3:=sbs(ss1, {seq(exprlst1[ll], ll=1.. cont}; ss4:=solve(ss3; Trans:=fff f lhs(fff=rhs(fff then lhs(fff ss5:=map(trans, ss4; rs2:=sbs(ss2, ss5; rs3:=rs1 mns rs2; Retrn(rS3; Retrn(rS1; Remarks Maple fncton dff(pexpr, x$n: comptes the n-th-order x-dervatve of pexpr. Maple fncton dsbs(derv=a, expr: performs dfferental sbstttons nto expressons. Maple fncton seq(f(, = m,...,n: generates the seqence f(m, f(m + 1,...,f(n. User s fncton getterms(pexpr: a short sbrotne whch s omtted here to fetch the monomals (wthot coeffcent from polynomal pexpr. User s fncton getmaxdffterm(pset: a short sbrotne whch s omtted here to fetch a term wth hghest order dervatve factor from the Pset Determne the Coeffcents In the above sbsecton, we gve a sbrotne to constrct the general form of conserved densty or flx for a gven rank. In ths sbsecton, we ntrodce another procedre GetConsLaw(EqSet :: set, P araset :: set, Rank :: posnt to determne the combnaton coeffcents. The man dea of ths sbrotne s based on Eq. (7. Once the rank of conserved densty s gven, the rank of the the assocated flx can be solved ot, namely the general form of conserved densty and ts assocated flx can be determned synchronosly. It s clear that the obtaned ρ and the correspondng J shold satsfy the defnton (2 for any solton of eqaton (1. By sng ths constrant, a compatblty condton can be obtaned. In detal, the procedre s as follows:

7 No. 1 An Extenson of Algorthm on Symbolc Comptatons of Conserved Denstes for Hgh-Dmensonal Nonlnear Systems 29 Let Rset = Weght(EqSet, ParaSet Let ρ = GenT erms(rset, Rank, tre Let J = GenTerms(Rset, Rank Ω(x + Ω(x, false Let Eq1 := ρ x + n =1 (J x Sbstttng EqSet nto Eq1, yeld Eq2 Collectng all the coeffcents of the prodct of potental fncton and ts dervatves, yelds a lnear system for combnaton coeffcents (C ρ, C J and the parameters of the eqaton. Solvng the lnear system, we obtan the solton sets. Then remove the trval soltons we obtan the conserved densty and assocated flx. 3 Applcatons 3.1 Example A: PCLaws of (2 + 1-Dmensonal BSE (11 The program gves the system wth free weght. If we take weghts of the varables as Ω( = 1, Ω(x = 1, Ω(y = 2, Ω(t = 4, we have the frst two densty-flx pars Rank 4: ( 2 x t + ( x x2 y 4 x xy x 2 xy 4 2 x y x + ( x x x x 2 y = 0, Rank5: ( xy t + ( x 2 y 2 y t 4 x y 2 4 xy y x 2 y y x xy 2 x + (4 2 x y + x 2 xy + 2 x 2 y y = 0. and the next for denstes are Rank 6: ρ 6 = 4 x x 2 2 x 2, Rank 7: ρ 7 = 4 x 2 y x 2 xy, Rank 8: ρ 8 = 15 2 x x x 2 x 3 2 x, 3 Rank 10: ρ 10 = 70 3 x x x 2 x x 3 x x x 2 x Example B: PCLaws of (2 + 1-Dmensonal KD Eqaton (9 Consder the (2+1-dmensonal KD eqaton (9 wth constant parameters α and β. It s clear that the system s not nform n rank nless Ω(β = 1. One shold call the program as the form of Weght({KD}, {β}, then the program otpts Ω( = 1, Ω(w = 2, Ω(x = 1, Ω(y = 2, Ω(t = 3, Ω(β = 1. Then we have the only frst two densty-flx pars wth no parameter constrant Rank 1: ( 3α 2 ( t x 2 3β 2 3αw x ( 3α w = 0, y Rank 2: ( 3α ( 2 2 t w 2 3α 2 w 4β x 2 x 2 ( + α 3 6w = 0. y 3.3 Example C: PCLaws of (3 + 1-Dmensonal Jmbo Mwa Eqaton For the (3 + 1-dmensonal Jmbo Mwa eqaton yt = 3 2 x z 1 2 x 3 y 3 2 x 2 y, (15 t s easy to fnd that the system s also wth free weght. If we take Ω( = 1, Ω(x = 1, Ω(y = 1, Ω(z = 3, Ω(t = 3, we obtan the only one densty-flx par ( xy t + ( x xy + y t 2 x y x x 2 y x + ( x x 2 x 2 2 x t xt y + ( x = 0. z 3.4 Example D: PCLaws of KP Eqaton For the (2 + 1-dmensonal KP eqaton xt = y x 6 x 2 x 4, (16 nfortnately, the program reports that the system has no PCLaws. However, f we take ts ntegraton form t = y 2 dx 6 x x 3, (17 and let v = y dx, yeld a copled eqaton t = v y 6 x x 3, y = v x. (18 Then for the PCLaws of Eq. (18, one can fnd t possesses two densty-flx pars Rank 2: ( t + (3 2 + x 2 x + ( v y = 0, Rank 4: ( 2 t + (4 3 + v x 2 2 x x + ( 2v y = Example E: PCLaws of (2 + 1-Dmensonal Shallow Water Wave System Consder the (2 + 1-dmensonal shallow water wave (SWW eqaton system t = x v y + 2αv hθ x 2 θh x, v t = v x vv y 2α hθ y 2 θh y, x

8 30 YANG X-Dong, RUAN Hang-Y, and LOU Sen-Ye Vol. 50 h t = h x h x hv y vh y, θ t = θ x vθ y, (19 where α s a constant parameter. We can fnd that the SWW system admts a Hamltonan formlaton [18] and nfntely many conservaton laws. Usng or package, t s easy to obtan the frst few PCLaws wth free α: Rank 1: ρ 1 = h, J 1 = { h, vh }, Rank 2: ρ 2 = hθ, J 2 = { hθ, vhθ }, Rank 3: ρ (1 3 = hθ 2, J (1 3 = { hθ 2, vhθ 2 }, ρ (2 3 = 2 h + v 2 h + h 2 θ, J (2 3 = { 3 h + v 2 h + 2h 2 θ, 2 vh + v 3 h + 2vh 2 θ }, ρ (3 3 = θ y vθ x + 2αθ, J (3 3 = { hθθ y v 2 θ y vθ x θ y vθv y, 2 } θ x hθθ x + vθ y + θ x + vθv x. However, the program of the drect method s faled to calclate the PCLaws nder Rank 4. The reason les n that the system possesses eght weghted varables or parameters. Then the general form of conserved densty and ts correspondng flx are natrally wth more ndetermned coeffcents. At last, t s waste of tme for a compter to solve sch a complcated over-determned systems wth more than 2015 eqatons and 84 varables whch are sed to determne the coeffcents. As the SWW eqaton s an evolton one, we admt Eler operator and homotopy operator (Operator Method to determne the coeffcents at the last step. Then we have the next few PCLaws. Rank 4: ρ (1 4 = hθ 3, J (1 4 = { hθ 3, vhθ 3}, ρ (2 4 = θθ y vθθ x + αθ 2, J (2 4 = { 1 4 (22 2v 2 + hθθθ y 1 4 θ3 h y vθθ x, 1 4 (22 2v 2 } hθθθ x θ3 h x + vθθ y, Rank 5: ρ (1 5 = hθ 4, J (1 5 = { hθ 4, vhθ 4}, ρ (2 5 = 3θ 2 θ y 3vθ 2 θ x + 2αθ 3, J (2 5 = { 3 10 (52 5v 2 + 2hθθ 2 θ y 3 5 θ4 h y 3vθ 2 θ x, 3 10 (52 5v 2 2hθθ 2 θ x θ4 h x + 3vθ 2 θ y }. 4 Smmary In ths paper, we present a new algorthm for symbolc comptatons of PCLaws for nonlnear systems and ts package n Maple. Usng or package, fve classcal nonlnear eqatons are sccessflly tested and the reslts are lsted. Unlke the (1 + 1-dmensonal case, we fnd that even for ntegrable models, there are few eqatons possesses polynomal conservaton laws. It s nterestng that f we ntrodce some stable potentals, say, v x = yy for the KP eqaton, we can fnd some PCLaws wth potentals. The work of comptng other knds of PCLaws ncldng more potentals s worthy of frther stdyng. On the other hand, we fnd that f one tres to fnd PCLaws n hgh ranks, one has to take very long tme to drectly solve the over-determned systems wthot sng Eler operator and homotopy operator when the gven systems wth more weghted varables and parameters. So, how to tlze the Eler operator and homotopy operator n solvng the PCLaws of the non-evolton systems s another mportant aspect, whch shold be consdered n the ftre work. Acknowledgment The athors are n debt to the helpfl dscssons wth Prof. R.X. Yao, Dr. M. Ja, and Dr. J.H. L. References [1] R.M. Mra, C.S. Gardner, and M.D. Krskal, J. Math. Phys. 9 ( [2] S.Y. Lo, J. Math. Phys. 35 ( [3] F. Magr, J. Math. Phys. 19 ( [4] V.E. Zakharov and L.D. Faddeev, Fnctonal Anal. Appl. 5 ( [5] C.S. Gardner, J.M. Greene, M.D. Krskal, and R.M. Mra, Commn. Pre Appl. Math. 27 ( [6] M.J. Ablowtz and H. Segr, Soltons and the Inverse Scatterng Transform, SIAM, Phladelpha (1981. [7] P.J. Olver, Applcaton of Le Grops to Dfferental Eqatons, Sprnger-Verlag, New York (1986. [8] X.D. Yang, H.Y. Ran, and S.Y. Lo, Commn. Theor. Phys. (Bejng, Chna 47 ( [9] M. Ito and F. Kako, Compt. Phys. Commn. 38 ( [10] M. Ito, Compt. Phys. Commn. 79 ( [11] Ü. Göktas and W. Hereman, M.Sc. Thess, Colorado School of Mnes, Golden, U.S.A. (1996. [12] A.V. Bocharov, DELA User Gde, Beaver Soft Programmng Team, New York (1991. [13] J.A. Sanders and J.P. Wang, Tech. Rep. 2, RIACA, Amsterdam (1994. [14] J.A. Sanders and J.P. Wang, Tech. Rep. WS-445, Vrje Unversty, Amsterdam (1995. [15] T. Wolf, A. Brand, and M. Mohammadzadeh, J. Symb. Comp. 27 ( [16] R.X. Yao and Z.B. L, Appl. Math. Compt. 173 ( [17] W. Hereman, Int. J. Qantm Chem. 106 ( [18] P. Dellar, Phys. Flds 15 (