Nonlocal Symmetries and Exact Solutions for PIB Equation

Transcription

1 Commun. Thor. Phys Vol. 58 No. 3 Spmbr Nonlocal Symmris and Exac Soluions for PIB Equaion XIN Xiang-Png 1 MIAO Qian 1 and CHEN Yong í 1 1 Shanghai Ky Laboraory of Trusworhy Compuing Eas China Normal Univrsiy Shanghai 0006 China Nonlinar Scinc Cnr and Dparmn of Mahmaics Ningbo Univrsiy Ningbo China Rcivd April 9 01 Absrac In his papr h symmry group of h +1-dimnsional Painlvé ingrabl Burgrs PIB quaions is sudid by mans of h classical symmry mhod. Ignoring h discussion of h infini-dimnsional subalgbra w consruc an opimal sysm of on-dimnsional group invarian soluions. Furhrmor by using h consrvaion laws of h rducd quaions w obain nonlocal symmris and xac soluions of h PIB quaions. PACS numbrs: 0.30.Jr 0.0.Hj 0.0.Sv Ky words: classical Li symmry mhod opimal sysm nonlocal symmry xplici soluion 1 Inroducion Symmry is on of h mos imporan concps in h ara of diffrnial quaions DEs spcially in parial diffrnial quaions PDEs. In h 19h cnury Sophus Li iniiad his sudis on coninuous groups Li group. H showd ha a poin symmry of a DE lads in h cas of an ODE o rducing h ordr of h DE irrspciv of any imposd iniial condiions and in h cas of a PDE o finding spcial soluions calld invarian soluions of h DE. Wih h dvlopmn of ingrabl sysms and solion hory h xnsions of Li s work o PDEs hav focusd on finding furhr applicaions of poin symmris o includ linarizaion mappings and soluions of boundary valu problms xnding h spacs of symmris of givn PDE sysm o includ local symmris [1 6] as wll as nonlocal symmris. [7 11] Local symmris admid by a PDE ar usful for finding invarian soluions. Ths soluions ar obaind by using group invarians o rduc h numbr of indpndn variabls. Local symmris admid by a nonlinar PDE ar also usful o discovr whhr or no h quaion can b linarizd by an invribl mapping and consruc an xplici linarizaion whn on xiss. An obvious limiaion of group-horic mhods is ha som PDEs of physical inrs possss fw symmris. Bu hy can admi nonlocal symmris whos infinisimal gnraors dpnd on ingrals of h dpndn variabls in som spcific mannr. Thr ar svral mhods o obain nonlocal symmris of givn PDE sysms. In a numbr of cass h nonlocal symmris may b asily obaind wih h hlp of a rcurrnc opraor. [1] Bu somims h rcurrnc opraors of givn sysm ar difficul o obain. Evn if on go h rcurrnc opraors of h sysm on also can no obain h nonlocal symmris. In Rf. [13] Akhaov and Gazizov providd a mhod for consrucing nonlocal symmris of DEs basd on h Li-Bäcklund hory. Morovr h fini symmry ransformaion and similar rducion canno b dircly applid o nonlocal symmris. Naurally i is ncssary o inquir as o whhr on can ransform nonlocal symmris ino Li poin symmris by xnding original sysm. In gnral nihr for h local nor for h nonlocal variabls h prolongaion dos no clos. Bluman inroducd h concp of ponial symmry or nonlocal symmry for a PDE sysm. W inroduc h concp in h 4h scion. Th ponial symmris of nonlinar PDEs hav bn sudid in h liraurs from svral diffrn poins of viw. Symmry group chniqus basd on local symmris provid on mhod for consrucing group invarian soluions of PDEs and linarizing nonlinar PDEs by invribl mappings. Local symmris of ponial sysm may yild nonlocal symmris of h givn sysm and h xisnc of nonlocal symmris lads o h consrucion of corrsponding invarian soluions as wll as o h linarizaion of nonlinar PDEs by non-invribl mappings. [14 16] In his papr aking h wll known +1-dimnsional Painlvé ingrabl Burgrs PIB quaions for a spcial xampl h PIB quaion was drivd from h gnralizd Painlvé ingrabiliy classificaion by Hong al. [17] u = uu y + avu x + bu yy + abu xx u x = v y 1 som xplicily xac soluions of h PIB quaion hav bn obaind via variabl sparaion approach [17 18] and mulipl Riccai quaions raional xpansion mhod. [19] W rsudy his quaion by using h nonlocal symmry dfind by Bluman s hory. Firs w us Olvr s Suppord by h Naional Naural Scinc Foundaion of China undr Gran No Innovaiv Rsarch Tam Program of h Naional Naural Scinc Foundaion of China undr Gran No and Shanghai Lading Acadmic Disciplin Projc undr Gran No. B41 ychn@si.cnu.du.cn c 011 Chins Physical Sociy and IOP Publishing Ld hp:// hp://cp.ip.ac.cn

2 33 Communicaions in Thorical Physics Vol. 58 mhod which only dpnds on fragmns of h hory of Li algbras o consruc h opimal sysm [0 3] of h +1-dimnsional PIB quaions hn us h consrvaion laws of h rducd quaions o consruc nonlocal symmris. This papr is arrangd as follows: In Sc. by using h classical Li symmry mhod w g h vcor filds of h PIB quaions 1. Thn h ransformaions laving h soluions invarian i.. is symmry groups ar obaind. In Sc. 3 wih h associad vcor filds obaind in Sc. w consruc h on-paramr opimal sysm of group-invarian soluions. Basd on h opimal sysm som rducions of Eq. 1 ar drivd. In Sc. 4 using h consrvaion laws of h rducd quaions w obain nonlocal symmris and xac soluions. Finally som conclusions and discussions ar givn in Sc. 5. Symmry Group of h PIB Equaion By applying h Li symmry mhod [4 6] w considr h on-paramr group of infinisimal ransformaions inx y u v of Eq. 1 givn by x = x + εxx y u v + oε y = y + εy x y u v + oε = + εtx y u v + oε u = u + εψx y u v + oε v = v + εφx y u v + oε whr ε is h group paramr. I is rquird ha Eqs. 1 b invarian undr h ransformaions and his yilds a sysm of ovrdrmind linar quaions for h infinisimals X Y T Ψ and Φ which can b solvd by virur of Mapl o giv Xx y u v = f + c 1 + c x Y x y u v = c 1y c 4 Tx y u v = c 1 + c + c 3 + c y + c 5 Ψx y u v = y uc 1 c u + c 4 Φx y u v= 1 d/df c 1 x c 1 +c av 3 a whr c 1 c c 3 c 4 c 5 ar arbirary consans and f is an arbirary funcion of. Hr w ak h f as 1 and for simpliciy. Th associad vcor filds for h on-paramr Li group of infinisimal ransformaions v 1 v v 3 v 4 v 5 v 6 v 7 ar givn as v 1 = x x + y y + y + u v = x x + y y + u u v v v 3 = v 4 = y + u v 5 = y u x + av v a v 6 = x v 7 = x 1 v. 4 a Equaions 4 show ha h following ransformaions dfind by xpεv i i = of variabls x y u v lav h soluions of Eqs. 1 invarian: xpεv 1 : x y u v x ε y ε ε xpεv : x y u v yε + u ε + av xε aε 1/ε x 1/ε y ε 1/ε u 1/ε v xpεv 3 : x y u v x y + ε u v xpεv 4 : x y u v x y ε u + ε v xpεv 5 : x y u v x y + ε u v xpεv 6 : x y u v x + ε y u v xpεv 7 : x y u v x + ε y u v + 1 a ε. Thn h following horm holds: Thorm 1 If { u = px y v = qx y is a soluion of Eqs. 1 hn so ar u 1 = x ε + p ε + y ε + ε + v 1 = x ε + q ε + y ε + ε + u = 1/ε p 1/ε x 1/ε y ε v = 1/ε q 1/ε x 1/ε y ε u 3 = px y ε u 4 = px y ε ε u 5 = px y ε u 6 = px ε y v 3 = qx y ε yε ε + xε aε + v 4 = qx y ε v 5 = qx y ε v 6 = qx ε y u 7 = px ε y v 7 = qx ε y 1 a ε. 3 Opimal Sysm and Rducions of PIB Equaions By using h gnraors v i of h Li-poin ransformaions in Eqs. 4 on can build up xac soluions of Eqs. 1 via h symmry rducion approach. This allows on o lowr h numbr of indpndn variabls of h sysm of diffrnial quaions undr considraion using h invarians associad wih a givn subgroup of h symmry group. In h following w prsn som rducions lading o xac soluions of h quaions of possibl physical inrs. Firsly w consruc an opimal sysm o classify h group-invarian soluions of Eqs. 1. As i is said in h Rf. [0] h problm of classifying group-invarian soluions rducs o h problm of classifying subgroups of h full symmry group undr conjugaion. And h problm of finding an opimal of subgroups is quivaln o ha of finding an opimal sysm of subalgbras. Hr

3 No. 3 Communicaions in Thorical Physics 333 by using h mhod prsnd in Rfs. [0 1] w consruc h opimal sysm of on-dimnsional subalgbras of Eqs. 1. Applying h commuaor opraors [v m v n ] = v m v n v n v m w g h commuaor abl lisd in Tabl 1 wih h i j-h nry indicaing [v i v j ]. And i follows Tabl 1 Li Brack. Li v 1 v v 3 v 4 v 5 v 6 v 7 v 1 0 v 1 v 0 1/v 4 1/v 7 0 v v 1 0 v 3 1/v 4 1/v 5 1/v 6 1/v 7 v 3 v v 3 0 v v 6 v 4 0 1/v 4 v v 5 1/v 4 1/v v 6 1/v 7 1/v v 7 0 1/v 7 v Proposiion Th opraors v i i = form a Li algbra which is a svn dimnsional symmry algbra. To compu h adjoin rprsnaion w us h Li sris in conjuncion wih h abov commuaor abl. Applying h formula Adxpεv i v j = v j ε[v i v j ] + 1/ε [v i [v i v j ]] and Tabl 1 on can hav h adjoin rprsnaion lisd in Tabl wih h i j-h nry indicaing Adxpεv i v j. Following Ovsiannikov [0] on calls wo subalgbras v and v 1 of a givn Li algbra quivaln if on can find an lmn g in h Li group so ha Adgv 1 = v ; whr Adg is h adjoin rprsnaion of g on v. Givn a nonzro vcor for xampl v = a 1 v 1 + a v + a 3 v 3 + a 4 v 4 + a 5 v 5 + a 6 v 6 + a 7 v 7. Our ask is o simplify as many of h cofficins a i as possibl hough judicious applicaions of adjoin maps o v. In his way omiing h daild compuaion on can g h following horm by h complicad compuaion: Tabl Adjoin Rprsnaion. Li v 1 v v 3 v 4 v 5 v 6 v 7 v 1 v 1 v + εv 1 v 3 + εv + ε /v 1 v 4 v 5 1/εv 4 v 6 + 1/εv 7 v 7 v ε v 1 v cosεv 3 + sinεv 3 1/ε v 4 1/ε v 5 1/ε v 6 1/ε v 7 v 3 v εv 3 v 3 v 4 + εv 5 v 5 v 6 v 7 εv 6 v 4 v 1 v + 1/εv 4 v 3 εv 5 v 4 v 5 v 6 v 7 v 5 v 1 + 1/εv 4 v 1/εv 5 v 3 v 4 v 5 v 6 v 7 v 6 v 1 + 1/εv 7 v 1/εv 6 v 3 v 4 v 5 v 6 v 7 v 7 v 1 v + 1/εv 7 v + εv 6 v 4 v 5 v 6 v 7 Thorm Th opraors gnra an opimal sysm S 1 givn by a1 v a 1 0 ; a v 3 + a 4 v 4 + a 7 v 7 a 1 = a = 0 a 3 0 ; b1 v 1 + a 3 v 3 + a 6 v 6 a 1 0 ; b αv 5 + a 7 v 7 a 1 = a = a 3 = 0 a 4 0 ; c1 v 1 + a 5 v 5 + a 6 v 6 a 1 0 ; c v 6 a 1 = a = a 3 = a 4 = a 5 = 0 a 6 0 ; d1 v 1 + a 5 v 5 a 1 0 ; d v 7 a 1 = a = a 3 = a 4 = a 5 = a 6 = 0 a 7 0 ; 1 v 1 a 1 0. W can s h cofficins of v i as No ha in h group of quivalnc ransformaions hr also includ discr ransformaions. Making us of S 1 w discuss h rducions and soluions of Eqs. 1 wih h cofficins as 1. Omiing h compuaion on can g h rducions in Tabl 3 using h gnral mhod. Tabl 3 Invarians and Rducion. Invarians paramrs Rducion a1 a b1 b u = y u = 1 Uξ η v = 1 V ξ η U + 1 ξu ξ + 1 ηuη + UUη + av U ξ + bu ηη + abu ξξ = 0 ξ = x η = y. U ξ V η = 0. u = Uξ η v = 1 a + V ξ η UUη + au ξv + bu ηη + abu ξξ = Uξη + ξ = ξ = x 1 η = y. U ξ V η = 0. v = 1 a x a V ξ η buηη + η + abu ξξ + au ξ V + UV η = 0 x η = y +. U ξ V η = 0. + u = Uξ η v = V ξ η x a ηuη + ηuu ξ av U ξ bηu ξξ abu ξξ η ξu ξ = 0 ξ = x y η =. U ξ + ηv ξ + η a = 0.

4 334 Communicaions in Thorical Physics Vol. 58 Tabl 3 Coninud c1 c d1 d 1 u = 1+y Uξ η v Invarians paramrs ξ = 1+x Rducion = 1+x a 1 a + 1 V ξ η UUη au ξv bu ηη abu ξξ = 0 η = 1+y. U ξ V η = 0. u = Uξ η v = V ξ η U η UU ξ bu ξξ = 0 ξ = y η =. V ξ = 0. u = +y + 1 Uξ η v = x a + 1 V ξ η UUη au ξv bu ηη abu ξξ = 0 ξ = x η = 1+y. U ξ V η = 0. u = Uξ η V = x aη + V ξ η Uη UU ξ bu ξξ = 0 ξ = y η =. V ξ = 0. u = y + 1 Uξ η v = x a + 1 V ξ η UUη + au ξv + buηη + abuξξ = 0 ξ = x η = y. U ξ V η = 0. 4 Nonlocal Symmris of h Rducd Equaions Th ponial symmry approach is an algorihmic procdur for sking nonlocal symmris of PDE sysms. This approach rquirs h xisnc of a consrvaion law of a givn sysm. Each consrvaion law allows h inroducion of on or mor auxiliary ponial variabls which ar nonlocally dfind wih rspc o h original dpndn variabls. Th rsuling ponial sysm yilds nonlocal symmris of h givn sysm if i admis local symmry gnraors ha do no projc ono local symmry gnraors of h givn sysm. In his scion w sk h nonlocal symmris of h rducd quaion insad of sking h nonlocal symmris of h PIB quaion dircly. Through h complx calculaion w can know ha hr ar no nonlocal symmris xcp h cas c and h cas d. As for h ohr cass on can obain a lo of xacly soluions using som ffciv mhod. Hr w omi h solving procss. For h cas c: U η UU ξ bu ξξ = 0 V ξ = 0 5 i is asy o g h soluion of h scond quaion of Eq. 5: V = V x + C. W sk h symmris of h firs quaion of h Eq. 5. In gnral on should obain consrvaion laws of his quaion firsly. For a PDE sysm nonrivial consrvaion laws aris from linar combinaions of h quaions of h PDE sysm wih muliplirs facors characrisics [7] ha yild nonrivial divrgnc xprssion. Thorm 3 A s of non-singular muliplirs {Λ σ x U U... l U} N σ=1 yilds a local consrvaion laws for a PDE sysm R{x; u} if and only if h s of idniis E U Λ σ x U U... l URx U U... k U 0 holds for arbirary funcion Ux whr E U is Eulr opraor wih rspc o U dfind as E U = U D i s D i1 D is + U i U i1 i s and D is oal drivaiv. Using his horm w sk muliplirs of h firs quaion of Eq. 5 considr zroh-ordr muliplirs Λξ η U firs ordr muliplirs Λξ η U U ξ and scond ordr muliplirs Λξ η U U ξ U ξξ rspcivly. Bu on only obains Λ = c whr c is an arbirary consan. So h consrvaion laws hav h form: D η U = D ξ 1/U + bu ξ. Th consrvaion laws yilds a pair of ponial quaion UW {ξ η; U W } givn by 1 W ξ = U W η = U + bu ξ 6 for som auxiliary ponial variabl W = Wξ η. In Eqs. 6 ponial variabl W is a nonlocal variabl i.. i canno b xprssd as a local funcion of giv variabl ξ η U and parial drivaivs of U. Th poin symmry maps any soluion of U W {ξ η; U W } o a soluion of UW {ξ η; U W }. Suppos U W {ξ η; U W } is invarian undr on-paramr Li group of poin ransformaions wih corrsponding infinisimal gnraor ṽ = αξ η U W δ δ + βξ η U W δξ δη + Aξ η U W δ δ + Bξ η U W δu δw. Dfiniion Th poin symmry of h ponial sysm UW {ξ η; U W } dfins a ponial symmry of Eq. 6 if and only if h infinisimals αξ η U W βξ η U W Aξ η U W dpnd xplicily on on or mor componns of W. Using Li group mhod on can obain h corrsponding infinisimal gnraor. Omiing h daild compuaion on can g h following rsuls by h complicad compuaion:

5 No. 3 Communicaions in Thorical Physics 335 Aξ η U W = 1 [b C 10 bu C10η W/b C 7 C 8 C 10ξ/ b b 3/ C10ξ/ b Bξ η U W = 1 b 3/ C 1 Uη + C U + C 1 ξ + C 4 C10ξ/ b C 9 C10η b C 10 + bu W/b C 7 ] ηb + 1 ξ C 1 C 4 ξ + C 6 + C 7 C10η b C10ξ/ C 8 + C 9 W/b C10ξ/ b αξ η U W = 1 C 1η + C ξ + C 4 η + C 5 βξ η U W = 1 C 1η + C η + C 3. Bas on h dfiniion h Eqs. 5 xis ponial symmris. Suppos C 1 = 0 C = 1 C 3 = 0 C 4 = 0 C 5 = 0 C 6 = 0 C 7 = 1 C 8 = 1 C 9 = 1 C 10 = 0 on can g Aξ η U W = 1 U W/b + b b Bξ η U W = 1/W/b αξ η U W = 1 ξ βξ η U W = η hn h vcor fild xprssion is ṽ = 1 ξ δ δξ + η δ δη 1 U 1/W/b + b δ b δu + δ 1/W/b δw.7 W us h sandard algorihm o sk xacly soluions of Eqs. 6 and h algorihm is omid hr. In ordr o obain invarian variabls w should solv h characrisic quaions w wri h corrsponding characrisic quaions in h form dξ = dη ξ η = bdu U 1/W/b + b = dw. 8 1/W/b Solving Eqs. 8 on can obain fx U = ηlnη + gx b W = bln X = ξ 9 lnη + gx η whr f g ar arbirary funcions of h corrsponding variabls. As a ranslad canonical coordina ˆX = lnξ which is a soluion of ṽ ˆX = 1. Variabls {X ˆX fx ˆX gx ˆX} ar canonical coordinas. In h ponial sysm U W {ξ η; U W } prform a local chang of variabls ξ η; U W X ˆX fx ˆX gx ˆX o obain a locally quivaln sysm Ũ W {X ˆX f g}. By subsiuing Eq. 9 ino Eqs. 6 w obain h rducion of Eqs. 6 Xg f = 0 f = bg. 10 In ordr o solv Eq. 10 on should subsiu h scond quaion ino h firs quaion of Eqs. 10 and obain h following rsul Xg bg = 0 11 sing g = h and Eq. 11 ransforma ino a firs ordr ordinary diffrnial quaion as 1 Xh bh = 0. Solving his quaion on can g h = X /4b 1 X /4b dx + b C 1 1 whr C 1 is ingral consan. In h Eq. 1 h ingraion X /4b dx can no b xprssd as linar combinaion of som lmnary funcion. Bu i can b xprssd as Error funcion i.. π rf 1/ 1/bX X /4b dx =. 1/b Error funcion is xprssd as rf θ = π θ So h soluions of Eqs. 10 ar 0 δ dδ. f = π rf 1/ 1/bX 1/b + C 1 X /4b X π rf 1/ 1/bX+ C1 1/b /4b g = b dx + C. 13 1/b Thus w can obain a nw soluion of Eq. 6 by subsiuing Eqs. 13 ino Eq. 9 U = π rf 1// 1/bX/ 1/b + C 1 X /4b ηlnη + [ π rf 1/ 1/bX + C1 1/b X /4b /b 1/b]dX + C W = bln whr X = ξ/ η. b lnη + [ π rf 1/ 1/bX + C 1 1/b X /4b /b 1/b]dX + C

6 336 Communicaions in Thorical Physics Vol. 58 And h nw soluion of h PIB quaion will b h following form: ux y = π rf 1/ 1/bX/ 1/b + C 1 X /4b ηlnη + [ π rf 1/ 1/bX + C1 1/b X /4b /b 1/b]dX + C vx y = V x + C 3 whr X = y/ η =. On h ohr hand w can discuss h nonlocally rlad subsysm of ponial sysm UW {ξ η; U W }. Th dpndn variabl U can b liminad from h ponial sysm UW {ξ η; U W } o yild h subsysm W {ξ η; W } givn by η W 1 ξ W b ξ W = 0 15 which is nonlocally rlad o h givn PDE bu locally rlad o h ponial sysm UW {ξ η; U W }. W solv h subsysm W {ξ η; W } and obain h soluions of PIB quaions. On can obain h corrsponding infinisimal gnraor by h complicad compuaion using h Li group mhod: ξc 1 η C 4η + C ξ δ ṽ = + C C δξ + 1 η + C η + C 3 δ δη 4C 8 C 10bη C 10ξ C 6 + C 7 m/b C 10ξ ηb + ξ /C 1 C 4ξ C 5 C 10ξ δ δw. 14 Suppos C 1 = 0 C = 1 C 3 = 0 C 4 = 1 C 5 = 0 C 6 = 0 C 7 = 0 C 9 = 0 C 10 = 0 on can obain: ṽ = η + 1 δ ξ δξ + η δ δη + ξ δ δw. 16 Solving h corrsponding characrisic quaions on can obain: W = ξ + η + FK 17 whr K = ξ + η/ η. By subsiuing Eq. 17 ino Eq. 15 w obain h rducion of Eq KF + 1 F + bf = If w s F = G hn Eq. 18 ransforms ino h following form: G = K b G 1 b G 19 on can s ha h Eq. 19 is Riccai yp quaion. By sing GK = bhk Eq. 19 bcoms h following form: H = H 1 KH. 0 b Thorm 4 If H is h soluion of h scond ordr ODE H + K/b H = 0 hn H / H is h soluion of h Riccai quaion 0. I is asy o solv h quaion H + K/b/ H = 0. Th soluion of h Eq. 0 has h form: H = K /4b K /4b dk. By using h formulas GK = bhk = F on can g : F = b K /4b K /4b dk and wih h hlp of h firs quaion of Eqs. 6 and h Eq. 17 on can obain U = + F K ξ i.. U = + b K /4b η K /4b dk. So w can g h nw soluion of PIB quaion: ux y = + b K /b η K /4b dk vx y = V x + C 4 1 whr K = y + / η =. 5 Conclusion and Discussion In his papr h symmris of h PIB quaions ar invsigad by mans of h classical Li symmry mhod. Th symmry algbra and group of Eqs. 1 ar obaind. Spcially h mos gnral onparamr groups of symmry ar givn ou as h composiion of ransforms in h svn various on-subgroups xpǫv 1... xpǫv 7. Nx h on-dimnsional subalgbras of h Li algbra of Eqs. 1 ar classifid. Finally h nonlocal symmris and nonlocal soluions of PIB quaions ar obaind. Using nonlocal symmris o consruc xplici soluions of PDEs is of considrabl inrs and valu. I would b possibl o xnd his approach o many ohr PDEs. Howvr hr is no a univrsal way o obain h nonlocal symmris. On can us various mhods o obain valuabl rsuls which is worhy of our furhr sudy.

7 No. 3 Communicaions in Thorical Physics 337 Rfrncs [1] S. Li Arch. Mah [] A.M. Wazwaz Compu. Mah. Appl [3] Z.Z. Dong and Y. Chn Commun. Thor. Phys [4] Y. Chn and Z.Z. Dong Nonlinar Anal [5] Z.Y. Yan Commun. Thor. Phys [6] X.R. Hu Y. Chn and L.J. Qian Commun. Thor. Phys [7] G.W. Bluman and S. Kumi Symmris and Diffrnial Equaions Springr-Vrlag Nw York [8] G.W. Bluman and A.F. Chviakov J. Mah. Phys [9] G.W. Bluman and G.I. Rid IMA J. Appl. Mah [10] X.R. Hu S.Y. Lou and Y. Chn Phys. Rv. E [11] S.Y. Lou X.R. Hu and Y. Chn J. Phys. A: Mah. Thor [1] G.A. Guhri J. Phys. A: Mah. Gn L905. [13] I.S. Akhaov and R.K. Gazizov J. Mah. Sci [14] S.Y. Lou and X.B. Hu J. Mah. Phys [15] A.F. Chviakov J. Mah. Phys [16] G.W. Bluman and A.F. Chviakov J. Mah. Anal. Appl [17] K.Z. Hong al. Commun. Thor. Phys [18] X.Y. Tang and S.Y. Lou Chin. Phys. L [19] Q. Wang Y. Chn and H.Q. Zhang Chaos Solion & Frac [0] L.V. Ovsiannikov Group Analysis of Diffrnial Equaions Nw York 198. [1] P.J. Olvr Applicaions of Li Groups o Diffrnial Equaions Springr-Vrlag Nw York [] N.H. Ibragimov and M. Torrisi Commun. Nonlinar Sci. Numr. Simul [3] M.L. Gandarias and N.H. Ibragimov Commun. Nonlinar Sci. Numr. Simul [4] S.Y. Lou and H.C. Ma J. Phys. A: Mah. Gn L19. [5] X.R. Hu and Y. Chn Commun. Thor. Phys [6] X.R. Hu and Y. Chn Commun. Thor. Phys