A Maple Package on Symbolic Computation of Conserved Densities for (1+1)-Dimensional Nonlinear Evolution Systems

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1 Commun. Theor. Phys. Bejng, Chna) ) pp c Internatonal Academc Publshers Vol. 47, No. 6, June 15, 2007 A Maple Package on Symbolc Computaton of Conserved Denstes for 1+1)-Dmensonal Nonlnear Evoluton Systems YANG Xu-Dong, 1 RUAN Hang-Yu, 1 and LOU Sen-Yue 1,2 1 Department of Physcs, Nngbo Unversty, Nngbo , Chna 2 Department of Physcs, Shangha Jao Tong Unversty, Shangha , Chna Receved October 16, 2006) Abstract A new algorthm for symbolc computaton of polynomal-type conserved denstes for nonlnear evoluton systems s presented. The algorthm s mplemented n Maple. The mproved algorthm s more effcent not only n removng the redundant terms of the general form of the conserved denstes but also n solvng the conserved denstes wth the assocated flux synchronously wthout usng Euler operator. Furthermore, the program conslaw.mpl can be used to determne the preferences for a gven parameterzed nonlnear evoluton systems. The code s tested on several well-known nonlnear evoluton equatons from the solton theory. PACS numbers: Jr, j, Wz Key words: conservaton laws, nonlnear evoluton systems, computer algebra 1 Introducton It s well known that many nonlnear evoluton systems, for example, the notable Koreteweg-de Vres KdV) equaton, possess varous remarkable mathematcal propertes such as nfntely many conservaton laws, [1] nfntely many symmetres, [2] nverse scatterng transformatons, [3,4] Bäcklund transformatons, Darboux transformatons, b-hamltonan structures, [5,6] etc. The exstence of nfntely many conserved denstes s an mportant lnk n the dscovery of the other specal propertes of the same gven equaton. Furthermore, t also predcts the ntegrablty. As a consequence, the work of the constructon of explct forms of conserved laws CLaws) plays an mportant role n the solton theory. Extensve studes of the CLaws can be found n Ref. [7], whch ncludes both Lagrangan and Hamltonan formulas. One may work out the frst few CLaws by hand. However, as ncreasng of the degree of conserved denstes, the constructon by hand becomes harder and harder and even mpossble. Recently, several symbolc computatonal algorthms for ths task have been presented and the correspondng programs have been developed on the dfferent platforms of symbolc computatonal systems. Ito et al. developed a REDUCE program CONSD [8] and the mproved verson SYMCD [9] by usng scalng propertes to compute polynomal conserved denstes. Bocharov and coworkers represented a package DELA [10] wrtten n Turbo PASCAL wth Le approach. Sanders et al. desgned a software package n Maple and FORM. [11,12] They use an extenson of the total dervatve operators to a Hesenberg algebra, whch allows them to nvert a total dervatve on ts mage. Goktas and Hereman also developed a package condens.m [13] n Mathematca, whch s based on the scalng propertes of the partal dfferental equatons PDEs), and ams to search for polynomal CLaws PCLaws) for nonlnear polynomal PDEs. Thomas Wolf developed a REDUCE package CONLAW1-4. [14] Hs approach s based on solvng over-determned system of PDEs, whch shows up n the computaton of Le-pont symmetres and generalzed symmetres. Yao et al. mproved the algorthm gven by Goktas and Hereman and desgned a maple package CONSLAW, [15] whch can be used to construct the PCLaws for polynomal PDEs automatcally. Recently, Hereman extended the algorthm nto mult-dmensons by usng Euler Operator and Homotopy Operator. [16] In ths paper, we ntroduce a new algorthm that s based n part on deas presented n Hereman [13,16] and Yao. [15] Our algorthm has the advantage that s fary straghtforward to mplement n any symbolc language. We also present a software package n Maple, whch automates the tedous computaton for the constructon of conserved denstes. The paper s organzed as follows. In Sec. 2, we outlne the algorthm and subroutnes. In Sec. 3, several examples are gven to demonstrate the applcatons. Then, a short summary s gven n Sec Computaton of Conserved Denstes 2.1 Defnton For smplcty, we consder a sngle nonlnear evoluton equaton u t = Hu, u x, u 2x,...), 1) The project supported by Natonal Natural Scence Foundatons of Chna under Grant Nos , , and , and the Natural Scence Foundaton of Zhejang Provnce

2 962 YANG Xu-Dong, RUAN Hang-Yu, and LOU Sen-Yue Vol. 47 where u = ux, t) s a potental functon and H s a polynomal of u and ts dervatve u x, u x = u/ x. A conservaton law s of the form ρ t + J x = 0, 2) whch s satsfed for all solutons of Eq. 1). The functonal ρx, t) s the conserved densty, Jx, t) s the assocated flux. As we know, the KdV equaton u t + 6uu x + u 3x = 0 3) possesses nfntely many conservaton laws and the frst three correspondng CLaws are u) t + 3u 2 + u 2x )x = 0, u 2 ) t + 4u 3 u 2 x + 2uu 2x ) x = 0, u 3 1 ) 9 2 u2 x + 2 u4 6uu 2 x + 3u 2 u 2x t u2 2x u x u 3x )x = 0. 4) It s clear that the conserved densty ρ = u, u 2, and u 3 u 2 x/2) are all nvarant under the symmetry transformaton x, t, u) λ 1 x, λ 3 t, λ 2 u), 5) where λ s a parameter. Scalng nvarance, whch s a specal Le-pont symmetry, s an ntrnsc property of many ntegrable nonlnear PDEs. Accordng to Eq. 5), wthout loss of the generalty, one can ntroduce the rank of varables ncludng the potental functon u and dependent varables x and t) as follows: Ωu) = 2, Ωx) = 1, Ωt) = 3. 6) Two clear and mportant results should be mentoned here. The frst one s that all monomals n Eq. 3) are unform n rank 5. The second one s that the frst three conserved denstes and the correspondng fluxes are unform n rank wth {2, 4, 6} and {4, 6, 8} respectvely. The ranks of the conserved denstes and the fluxes satsfy ΩJ) = Ωρ) + Ωx) Ωt), 7) where Ωρ) and ΩJ) are the rank of the conserved denstes and the assocated fluxes. Unform n rank plays an mportant role n our algorthm. Our algorthm explots ths dea n the constructon of conserved denstes. 2.2 Algorthm The computer algebrac algorthm and routnes to construct the CLaws for Eq. 1) are descrbed as n the followng steps Determne the Weghts of Varables and Parameters The work of determnng the weghts of varables and parameters can be easly done by the defnton of unform n rank. In our algorthm, we defne the weght of partal dervatves as follows: u ) u ) Ωu x ) = Ω x = Ω x = Ωu) Ωx), 8) where u denotes arbtrary potental functon and x s one of the dependent varable. For example, the ranks of three terms n Eq. 3) are Ωu) Ωt), 2Ωu) Ωx), Ωu) 3Ωx), respectvely. The requrement of unformty n rank leads to Ωu) Ωt) = 2Ωu) Ωx) = Ωu) 3Ωx) wth the soluton {Ωu) = 2Ωx), Ωt) = 3Ωx)}, where Ωx) can be an arbtrary constant. In our algorthm, lettng Ωx) = 1, one can drectly obtan a general rank soluton 6). However, there ndeed exsts such a case that the equaton s not unform n rank. Fortunately, we can use such a trck gven by Hereman by ntroducng some auxlary parameters wth unknown) weghts. The detals are shown n Ref. [13]. Here, we construct such a subroutne as W eght EqSet :: set, P araset :: set), where EqSet denotes a set of equatons and ParaSet s a set of parameters wth unknown weghts. Takng the KdV-mKdV equaton u t + αuu x + βu 2 u x + u 3x = 0 as a smple example, by settng the parameter α possesses an unknown weght, one should call the subroutne as W eght {u t = αuu x βu 2 u x u 3x }, {α}). In detal, the procedure proceeds as follows: Collectng varables of EqSet and parameters n P araset, we derve the set S0 = {u, x, t, α}; Gatherng all the monomals of the equaton, we derve the set S1 = {u t, αuu x, βu 2 u x, u 3x }; Selectng all the partal dervatves of the equaton, we let the set S2 = {u t, u x, u 3x }; Replacng each element n S2 wth formula u/ x u/ x = u/x, we have the set S3 = {u t = u/t, u x = u/x, u 3x = u/x 3 }; Substtutng S3 nto S1, we derve a new set S4 = {u/t, αu 2 /x, βu 3 /x, u/x 3 }; Replacng each element n S4 wth formula f = lnf), we have the set S5 = {lnu) lnt), lnα) + 2 lnu) lnx), lnβ) + 3 lnu) lnx), lnu) 3 lnx)};

3 No. 6 A Maple Package on Symbolc Computaton of Conserved Denstes for 1+1)-Dmensonal Nonlnear Evoluton Systems 963 Removng all the monomals wthout weght for each element n S5, we have the result S6 = {lnu) lnt), lnα)+ 2 lnu) lnx), 3 lnu) lnx), lnu) 3 lnx)}; Replacng each element n S0 wth formula f f = expω f ), where Ω f denotes not partal dervatve but an element of array Ω, then we have the set S7 = {u = expω u ), x = expω x ), t = expω t ), α = expω α )}; Substtutng S7 nto S6, we derve a new set S8 = {Ω u Ω t, Ω α + 2Ω u Ω x, 3Ω u Ω x, Ω u 3Ω x }; Constructng the rank equaton wth the condton of unform n rank, yelds S9 = {Ω u Ω t = Ω α + 2Ω u Ω x, Ω u Ω t = 3Ω u Ω x, Ω u Ω t = Ω u 3Ω x }; Addng the general assumpton Ω x = 1 to set S9, we derve the set S10 = {Ω u Ω t = Ω α +2Ω u Ω x, Ω u Ω t = 3Ω u Ω x, Ω u Ω t = Ω u 3Ω x, Ω x = 1}; Solvng equaton S10, we get the soluton Sol = {Ω u = 1, Ω x = 1, Ω t = 3, Ω α = 1}. Accordng to Sol, four cases should be consdered, a) Sol s NULL, namely the equaton s not unform n rank, then t prompts the user wth nformaton and ends the process. b) Sol wth free weght, whch prompts the user to nput a value, then go on. c) Sol wth fractal weght, then one should multply each weght n Sol wth the lease common multple of the denomnators n the fractal weghts, then go on. d) Otherwse, go on. Selectng functons and parameters wth postve weght from Sol, we have the set S11 = {Ω u = 1, Ω α = 1}. Selectng dependent varables wth negatve weght from Sol and transformng weght to postve, we derve the set S12 = {Ω x = 1, Ω t = 3}; Unon set S11 and S12, then remove the tem of Ω t and convert nto the form as Rset = {[u, 1], [α, 1], [x, 1]}; Convert each element n set S12 nto a lst as the form of [M, Rankt) RankM)], where M s a varable and RankM) s the correspondng weght. Then collectng the obtaned lst nto a set yelds V set = {[x, 2], [t, 0]}. Rset and V set are the fnal result whch wll be used as the key parameters n the next subroutne n constructng the general form of conserved denstes Construct the General Form of Conserved Denstes and Flux It s clear that the every monomals n ) P T ) nx W m nx are unform n rank [m ΩW )] + n Ωx), 9) where W s a potental functon or a weghted parameter, and ΩX) denotes the weght of the assocated varable X. In the above subsecton, the weghts of varables and parameters are obtaned as the set Rset. Now, we utlze Rset to construct the general form of conserved denstes. For a gven rank Rank, the procedure GenT ermsrset :: set, Rank :: posnt, Remove :: boolean) wll automatcally fnd all the possble combnatons nto a set as the form of {[m, n]}, whch satsfes [m ΩW )] + n Ωx) = Rank. Takng Rset = {[α, 1], [x, 1], [u, 2]} and Rank = 2 for example, n detal, the procedure proceeds as follows: Step 1 Construct all the possble combnatons nto the set {[m, n]}. Sort the Rset on rank n descendng order then convert nto two lsts as the form L1 = [V 1, V 2,..., V n ] and L2 = [ΩV 1 ), ΩV 2 ),..., ΩV n )], we have two lsts L1 = [u, α, x] and L2 = [2, 1, 1]; For each element n L1, f V s a potental functon or a weghted parameter then append ts address n L1 nto lst L3, otherwse nto lst L4, we derve L3 = [1, 2] and L4 = [3]; Set S1 = {[Rank, [ ]]};

4 964 YANG Xu-Dong, RUAN Hang-Yu, and LOU Sen-Yue Vol. 47 for from 1 to nopsl2) do Set S2 = {} Set vrank = op,l2) for jj from 1 to nopss1) do Set L5 = opjj,s1) Set rrank = op1,l5) Set L6 = op2,l5) for kk from 0 to rrank/vrank do Set S2 = S2 unon {[rrank kk vrank, [opl6),kk]} od od Set S1 = S2 od Remarks Maple functon nopsx): the number of elements of a lst or set X. Maple functon opx): return all elements of a lst or set X. Maple functon opn,x): return the n-th element of a lst or set X. Now, we have the set S1 = {[1, [0, 1, 0]], [0, [0, 1, 1]], [0, [0, 0, 2]], [0, [0, 2, 0]], [0, [1, 0, 0]], [2, [0, 0, 0]], [1, [0, 0, 1]]} for each element [rrank, rlst] n set S1, f rrank = 0 then nsert rlst nto S3, else omt t. From now on, we get all possble combnatons {[m, n]} as the result S3 = {[1, 0, 0], [0, 0, 2], [0, 2, 0], [0, 1, 1]}. In constructng the general form of densty ρ, some redundant terms whch can be wrtten as a total dervatve to spatal varable x should be removed as the result belong to the same equvalence class. However, for flux J the work of removng terms s not necessary. One should call the GenT erms wth Remove = true for densty ρ or Remove = false for flux J. Step 2 Remove redundant terms whch can be wrtten as a total dervatve. Frstly, we take an example to llustrate the man dea of our algorthm. If take RSet = {[u, 2], [x, 1]} and Rank = 8, one can easly obtan the correspondng L1 = [u, x], L2 = [2, 1], L3 = [1], L4 = [2] and S3 = {[4, 0], [3, 2], [2, 4], [1, 6], [0, 8]} respectvely. Then usng formula 9), we have the results u 4 ) 0x = u 4, u 3 ) 2x = 6uu 2 x + 3u 2 u 2x, u 2 ) 4x = 6u 2 2x + 8u x u 3x + 2uu 4x, u 1 ) 6x = u 6x, 10) and the constant term u 0 ) 8x = 0 should be omtted. Gatherng all the monomals on the rght hand of equaton 10), we derve the set rs1 = {u 4, uu 2 x, u 2 u 2x, u 2 2x, u x u 3x, uu 4x, u 6x }. Naturally, the form of the conserved flux of rank 8 can be wrtten as J = C J1 u 4 + C J2 uu 2 x + C J3 u 2 u 2x + C J4 u 2 x + C J5 u x u 3x + C J6 uu 4x + C J7 u 6x. Now we transform the terms for n 0) n 10) nto the form u 3 ) 2x = u 3 ) 1x ) x = 3u 2 u x ) x, u 2 ) 4x = u 2 ) 3x ) x = 6u x u 2x + 2uu 3x ) x = 6u x u 2x ) x + 2uu 3x ) x, u 1 ) 6x = u 1 ) 5x ) x = u 5x ) x. 11) Then expandng Eq. 11) yelds 2u 2 u x ) x = {6uu 2 x + 3u 2 u 2x }, 6u x u 2x ) x = {6u 2 2x + 6u x u 3x }, 2uu 3x ) x = {2u x u 3x + 2uu 4x }, u 5x ) x = {u 6x }. 12) It s easy to fnd that the monomals group comng from each { } n Eq. 12) can be constructed to a total dervatve by a proper coeffcent combnaton, namely each term can be replaced by the other terms n the same group namely a same equvalence class). Now, the only thng left us to do s to remove a proper term n each group. One of the most effcent way s to remove the term whch possesses the hghest dervatve order n each group. One can easly get the remove terms as the set rs2 = {u 2 u 2x, u x u 3x, uu 4x, u 6x }. Naturally, we have the result rs3 = rs1 mnus rs2 = {u 4, uu 2 x, u 2 2x} and thus the form of the conserved densty of rank 8 s gven as ρ = C ρ1 u 4 + C ρ2 uu 2 x + C ρ3 u 2 2x. The advantage of our algorthm s drectly n computaton and success n avodng the ntegral operaton whch s a weakness of the mostly symbolc computaton systems. Its algorthm s descrbed as follows: for from 1 to nopss3) do L7: = op, S3); T : = 1; for jj from 1 to nopsl3) do Compute T = od; ll: = opjj, L3); T : = T opll, L1) ˆ opll, L7); f T <> 1 then ll: = op1,l4); W m

5 No. 6 A Maple Package on Symbolc Computaton of Conserved Denstes for 1+1)-Dmensonal Nonlnear Evoluton Systems 965 var: = opll,l1); dervatve varable nn: = opll,l7); dervatve order f nn = 0 then Compute P = T ) nx P: = T ; else P: = dfft, var$nn); rs1[]: = gettermsp); f Remove then Compute the remove term f nn = 0 then P: = 0; elf nn = 1 then P: = T ; elf nn > 1 then P: = dfft,var$nn-1)); S4: = gettermsp); for jj from 1 to nopss4) do Term[jj]: = getmaxdfftermdffopjj,s4),var)); od; rs2[]: = {seqterm[ll]),ll = 1..nopsS4))}; rs3[]: = rs1[] mnus rs2[]; od; else rs1[]: = {}; rs3[]: = {}; f Remove then else Return{seqoprS3[ll]),ll = 1..nopsS3))}); Return{seqoprS1[ll]),ll = 1..nopsS3))}); Remarks Maple functon dffpexpr,x$n): computes the n-thorder x-dervatve of pexpr. Maple functon seqf), = m,..., n): generates the sequence fm), fm + 1),..., fn). User s functon gettermspexpr): a short subroutne whch s omtted here to fetch the monomals wthout coeffcent) from polynomal pexpr. User s functon getmaxdfftermpset): a short subroutne whch s omtted here to fetch a term wth hghest dervatve order factor from the pset. For a sngle nonlnear evoluton equaton, the algorthm to remove the redundant terms can be smply descrbed as follows: Compute P = N 1 =1 T = N N2 ) 1 =1 j=1 W mj j, where N 1 = nopss3), N 2 = nopsl3), W j = opopj, L3), L1), n x m j = opopj, L3), op, S3), n = opop1, L4), op, S3)) and x = opop1, L4), L1); Fetch all the monomals wthout coeffcent) from polynomal P nto a set rs1; For each element M1 n set rs1, fetch the factor wth the hghest dervatve order nto M2. If the power of M2 s equal to 1 then unon M1 nto rs2; Set rs3 = rs1 mnus rs Determne the Coeffcents In the above subsecton, we gve a subroutne to construct the general form of conserved densty or flux for a gven rank. In ths subsecton, we ntroduce another procedure GetConsLawEqSet :: set, P araset :: set, Rank :: posnt) to determne the combnaton coeffcents. The man dea of ths subroutne s based on Eq. 7). Once the rank of a conserved densty s gven, the rank of the assocated flux can be solved out, namely the general form of conserved densty and ts assocated flux can be determned synchronously. It s clear that the obtaned ρ and the correspondng J should satsfy the defnton 2) for any soluton of equaton 1). By usng ths constrant, a compatblty condtons can be obtaned. The detals of the procedure are as follows: Let Rset = W eghteqset, P araset); Let ρ = GenT ermsrset, Rank, true); Let J = GenT ermsrset, Rank Ωx) + Ωt), false); Let Eq1 = ρ t + J x ;

6 966 YANG Xu-Dong, RUAN Hang-Yu, and LOU Sen-Yue Vol. 47 Substtutng EqSet nto Eq1 yelds Eq2; Collectng all the coeffcents of the products of potental functons and ther dervatves yelds a lnear system for combnaton coeffcents C ρ,c J ) and the model parameters; Solvng the lnear system, we obtan the soluton sets. conserved denstes and assocated fluxes. Then removng the trval solutons, we obtan the The drect solvng of the over-determned algebrac equatons drect method) s a fary straghtforward method n the determnaton of the conserved densty-flux par synchronously. However, as the ncreasng of the degree of conserved densty, the work to solvng the over-determned systems becomes harder and harder and even mpossble or waste of tme. The algorthm of usng Euler operator and homotopy operator operator method) gven by Hereman may be a canddate way to determne the coeffcents, detals are shown n Ref. [16]. Then we have two packages n Maple, one s by drect method, the other s by operator method. 3 Applcatons 3.1 Example A The frst example we consder s the ffth-order KdV equaton u t + uu 3x + αu x u 2x + βu 2 u x + u 5x = 0 13) wth constant parameters α and β. Especally, for α = 2, β = 3/10, equaton 13) reduces to the usual ffth-order KdV equaton. The SK equaton, due to Sawada and Kotera, can be obtaned for α = 5/2, β = 1/5. The KK equaton, due to Kaup and Kupershmdt, corresponds to α = 1, β = 1/5 and the equaton due to Ito arses for α = 2, β = 2/9. Now, we study ts PCLaws. It s clear that the equaton s unform n rank. Frstly, the program gves the weght of the varables Ωu) = 2, Ωx) = 1, Ωt) = 5. Then we have the followng results: Rank 2 There are no constrants on the parameters. One can easly rewrte Eq. 13) nto the form of u) t + u 4x + uu 2x + α 1 u 2 x + β 2 3 u3) = 0. x Rank 4 Wth the parameter constrant α = 2, the conserved densty-flux par can be wrtten as u 2 ) t + 2uu 4x 2u x u 3x + u 2 2x + 2u 2 u 2x + β 2 u4) = 0. x In the above two results, we gve the result as densty-flux par. Because the forms of the fluxes become more and more lengthy as the ncreasng of the rank, for the next few results, we only lst the denstes and ts parameter constrants. Rank 6 For β = α 2 /5) + 7α/10) 3/10), α free, ρ 6 = 15u 2 x + 1 2α)u 3. Rank 8 For α = 2, β free, ρ 8 = 6u 2 2x 6uu 2 x + βu 4. For β = 2α 2 /45) + 7α/45) + 4/45), α free, ρ 8 = 675u 2 2x 1352α + 1)uu 2 x + 2α + 1) 2 u 4. Rank 10 For α = 2, β = 3/10, ρ 10 = 500u 2 3x + 700uu 2 2x 350u 2 u 2 x + 7u 5. The runnng tmes from the two packages are lsted n Table 1. Table 1 Eq. 13). The runnng tmes from the two packages for Rank Terms of Densty Operator Method Drect Method Example B We take the KdV-mKdV equaton, u t + αuu x + βu 2 u x + u 3x = 0, 14) wth constant parameters α and β as the second llustraton example. Because the equaton s obvously not unform n rank under the case that no parameter wth weght, one should call the program as the example whch s shown n the subsubsecton Frstly, the program gves the weght of the varables Ωu) = 1, Ωx) = 1, Ωt) = 3, Ωα) = 1. It s easy to fnd that the frst few conserved denstes are all wth no parameter constrants. The frst two denstyflux pars are Rank 1 Rank 2 u 2 ) t + u) t + u 2x + α 2 u2 + β 3 u3) = 0, x 2uu 2x u 2 x + 2α 3 u3 + β 2 u4) x = 0, and the next three conserved denstes are

7 No. 6 A Maple Package on Symbolc Computaton of Conserved Denstes for 1+1)-Dmensonal Nonlnear Evoluton Systems 967 Rank 4 Rank 6 ρ 4 = 6u 2 x + 2αu 3 + βu 4, ρ 6 = 36u 2 2x 60uα + βu)u 2 x + 5α 2 u 4 + 6αβu 5 + 2β 2 u 6, Rank 8 ρ 8 = 216u 2 3x + 504uα + βu)u 2 2x 252βu 4 x 420u 2 α + βu) 2 u 2 x + u 5 5β 3 u αβ 2 u α 2 βu + 14α 3 ). The calculatng tmes of the two packages are lsted n Table 2. Table 2 Eq. 14). The calculatng tmes of the two packages for Rank Terms of Densty Operator Method Drect Method Example C Thrdly, we consder the classcal Drnfel d Sokolov Wlson equaton, u t + pvv x = 0, v t + qv 3x + ruv x + svu x = 0, 15) where p, q, r, and s are arbtrary parameters. One can fnd that equaton 15) s unform n the rank even f the parameter wth no weght. Frstly, the program reports the weght of the varables Ωu) = 2, Ωv) = 2, Ωx) = 1, Ωt) = 3. Next, we lst some of the results as follows: Rank 2 Two sets of conserved densty-flux pars are gven as u) t pv2) x = 0, for no parameter constrant, and v) t + qv 2x + suv) x = 0 for r = s. One can fnd that the result s not smlar to the one gven by Yao n Ref. [15]. Indeed, n the case of the parameter constrant for r = s, equaton 15) possesses two basc solutons of conserved densty u and v. However, one of the solutons u should be removed from the results because t s a specal case for no parameter constrant. Rank 4 Under no parameter constrant, one set of conserved densty-flux par s [r 2s)u 2 pv 2 ] t + 2pqvv 2x + pqv 2 x 2psuv 2 ) x = 0. Rank 6 Under no parameter constrant, ρ 6 = 3qr s)u 2 x 9pqv 2 x + 3pr + 2s)uv 2 2r s)r + 2s)u 3. Rank 8 Under parameter constrant r = 2s, ρ 8 = 3q 2 su 2 2x 36qs 2 uu 2 x + 60pqsvu x v x + 72pqsuv 2 x 27pq 2 v 2 2x + 16s 3 u 4 6sp 2 v 4 24ps 2 u 2 v 2. The runnng tmes of the two packages are lsted n Table 3. Table 3 The runnng tmes of the two packages for Eq. 15). Rank Terms of Densty Operator Method Drect Method Example D Fnally, we consder the CLaws of a type of varant Boussnesq equaton u t + v x + uu x + su xx = 0, v t + uv) x + rv xx + pu xxx = 0, 16) where p, r, and s are arbtrary constant parameters. In the same way, the program frst outputs Ωu) = 1, Ωv) = 2, Ωx) = 1, Ωt) = 2. Then, we have the frst two densty-flux pars wth no parameter constrants. Rank 1 Rank 2 u) t + v + 1 ) 2 u2 + su x = 0, x v) t + rv x + pu 2x + uv) x = 0, The next three conserved denstes wth parameter constrant r = s read: Rank 3 Rank 4 Rank 5 ρ 3 = uv ; ρ 4 = pu 2 x 2su x v v 2 u 2 v ; ρ 5 = 3puu 2 x + 4p + s 2 )u x v x 6suvu x u 3 v 3uv 2. The calculatng tmes of the two packages are lsted for Eq. 16).

8 968 YANG Xu-Dong, RUAN Hang-Yu, and LOU Sen-Yue Vol. 47 Table 4 Eq. 16). The calculaton tmes of the two packages for Rank Terms of Densty Operator Method Drect Method Summary In ths paper, we present a new algorthm for symbolc computaton of PCLaws of nonlnear evoluton systems and ts package n Maple. Usng our package, four classcal nonlnear evoluton equatons are successfully tested and the results are lsted. The mproved algorthm s more effcent n removng the redundant terms of the general form of the conserved denstes. From the four tables of runnng tme, one can fnd that the drect method s more suffcent under small rank, whle the operator method has ts advantage wth large rank. Importantly, the algorthm of drect method can be extended to solve the PCLaws of the PDEs wth more than one spatal varable. However, the way of removng a proper term n each equvalent class should be modfed. Acknowledgment The authors are n debt to the helpful dscussons wth Prof. R.X. Yao, Drs. X.Y. Tang, M. Ja, Y. Chen, and H.C. Hu. References [1] R.M. Mura, C.S. Gardner, and M.D. Kruskal, J. Math. Phys ) [2] S.Y. Lou, J. Math. Phys ) [3] C.S. Gardner, J.M. Greene, M.D. Kruskal, and R.M. Mura, Commun. Pure Appl. Math ) 97. [4] M.J. Ablowtz and H. Segur, Soltons and the Inverse Scatterng Transform, SIAM, Phladelpha 1981). [5] F. Magr, J. Math. Phys ) [6] V.E. Zakharov and L.D. Faddeev, Functonal Anal. Appl ) 280. [7] P.J. Olver, Applcaton of Le Groups to Dfferental Equatons, Sprnger-Verlag, New York 1986). [8] M. Ito and F. Kako, Comput. Phys. Commun ) 415. [9] M. Ito, Comput. Phys. Commun ) 547. [10] A.V. Bocharov, DELA User Gude, Beaver Soft Programmng Team, New York 1991). [11] J.A. Sanders and J.P. Wang, Tech. Rep. 2, RIACA, Amsterdam 1994). [12] J.A. Sanders and J.P. Wang, Tech. Rep. WS-445, Vrje Unverstet, Amsterdam 1995). [13] Ü. Göktas and W. Hereman, M.Sc. thess, Colorado School of Mnes, Golden, CO, U.S.A, 1996). [14] T. Wolf, A. Brand, and M. Mohammadzadeh, J. Symb. Comp ) 221. [15] R.X. Yao and Z.B. L, Appl. Math. Comput ) 616. [16] W. Hereman, Int. J. Quantum Chem ) 278.