NEW EXACT ANALYTICAL SOLUTIONS FOR THE GENERAL KDV EQUATION WITH VARIABLE COEFFICIENTS. Jiangsu, PR China

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1 athematcal and Computatonal Applcatons Vol. 19 No. pp NEW EXACT ANALYTICAL SOLUTIONS FOR THE GENERAL KDV EQUATION WITH VARIABLE COEFFICIENTS Bao-Jan Hong 1 and Dan-Chen Lu 1* 1 Faculty of Scence Jangsu Unversty 11 Zhenjang Jangsu PR Chna Department of Basc Courses Nanjng Insttute of Technology 11167Nanjng Jangsu PR Chna ludanchenujs@16.com dclu@ujs.edu.cn Abstract- In ths paper a general algebrac method based on the generalzed Jacob ellptc functons expanson method the mproved general mappng deformaton method and the extended auxlary functon method wth computerzed symbolc computaton s proposed to construct more new exact solutons of a generalzed KdV equaton wth varable coeffcents. As a result eght famles of new generalzed Jacob ellptc functon wave solutons and Weerstrass ellptc functon solutons of the equaton are obtaned by usng ths method some of these solutons are degenerated to solton-lke solutons trgonometrc functon solutons n the lmt cases when the modulus of the Jacob ellptc functons m 1 or whch shows that the general method s more powerful and wll be used n further works to establsh more entrely new solutons for other knds of nonlnear partal dfferental equatons arsng n mathematcal physcs. Key words- Generalzed KdV equaton wth varable coeffcents; general algebrac method; exact solutons; generalzed Jacob ellptc functon wave-lke solutons 1. INTRODUCTION Nonlnear partal dfferental equatons (NLPDEs) are wdely used to descrbe complex physcal phenomena arsng n the world around us and varous felds of scence. The nvestgaton of exact solutons of NLPDEs plays an mportant role n the study of these phenomena such as the nonlnear dynamcs and the mechansm behnd the phenomena. Wth the development of solton theory many powerful methods for obtanng exact solutons of NLPDEs have been presented such as nverse scatterng transformaton [1] Hrota blnear method [] Bäcklund transformaton [] Darboux transformaton [] homotopy perturbaton method [] extended Rccat equaton ratonal expanson method [6] asymptotc methods [7] extended auxlary functon method [8] algebrac method [9] Jacob ellptc functon expanson method [1]and so on [11-1]. In [1][1] Hong proposed a generalzed Jacob ellptc functons expanson method to obtan generalzed exact solutons of NLPDEs. In [16] Hong et al. proposed an

2 B.-J. Hong and D.-C. Lu 19 mproved general mappng deformaton method to obtan generalzed exact solutons of the general KdV equaton wth varable coeffcents (GVKDV). Whch s more general than many other algebra expanson methods [68-1] etc. The soluton procedure of ths method by the help of atlab or athematca s of the utmost smplcty and ths method can be easly extended to all knds of NLPDEs. In ths work we wll proposed the general algebrac method to obtan several new famles of exact solutons for the GVKDV equatons. The rest of ths paper s organzed as follows. In secton we brey descrbe the new general algebrac method. In secton several famles of solutons for the GVKdV equaton are obtaned some of whch are degenerated to new soltary-lke solutons and new trangular-lke functons solutons n the lmt case. In secton some conclusons are gven.. SUARY OF THE GENERAL ALGEBRAIC ETHOD Consder a gven nonlnear evoluton equaton wth one physcal feld u( x t) n two varables x and t P( u u u u ). (1) t x xx We seek the followng formal solutons of the gven system by a new ntermedate transformaton: n 1. () n n u( ) A ( t) ( ) A ( t) ( ) Where At () An () t are functons of t to be determned later. ( xt ) are arbtrary functons wth the varables x and t. The parameter n can be determned by balancng the hghest order dervatve terms wth the nonlnear terms n Eq.(1). And ( ) s a soluton of the followng ordnary dfferental equaton (ODE) ' ( ) a ( t) ( ). () Substtutng Eqs. () and () nto Eq. (1) and settng the coeffcents of ( )( 1 ) and s j x ( ) a ( t) ( )( s 1; j 11 ) to zero yeld a set of algebrac equatons for At () An () t and. Usng the athematca to solve the algebrac

3 196 New Exact Analytcal Solutons for the General Kdv Equaton equatons and substtutng each of the solutons of the set.e. each of the expressons of ( ) nto Eq. () we can get the solutons of Eq. (1). In order to obtan some new general solutons of Eq.()we assume that () have the followng solutons: ( ) c c e( ) c f ( ) c g( ) c h( ). () 1 Where c c ( t)( ) are functons of t to be determned later the four functons e e( ) f f ( ) g g( ) h h( ) are expressed as the follows: 1 F F F ' p qf rf lf p qf rf lf p qf rf lf p qf rf lf e f g h.() ' ' ' ' Where p q r l are arbtrary constants whch ensure denomnator unequal to zero so do the followng stuatons and F F( ) s a soluton of the followng ODE ' F A BF CF DF EF F '' BF CF D EF. (6) Where ' denotes d d '' denotes d d A B C D E are arbtrary constants so do the followng stuatons the four functons e f g h satsfy the followng relatons: e' qeh rfh l( De Bef Cfg Ef ) f ' peh rgh l( Ae Def Cg Efg) g ' qgh pfh l(aef Df Bfg Eg ) h' ( Dp Aq) e ( Bp Dq Ar) ef ( Cp Eq Br) fg ( Ep Dr) f ( Cq Er) g f eg h Ae Bf Cg Def Efg pe qf rg lh 1 And e f g h satsfy one of the followng relatons at the same tme. Famly 1:When p.(7) ( Cl r ) h C Clh Br(1 lh qf ) e Ae r Dr ef (Cq Er) f (ler Clq) fh ( Eqr Cq ) f.(7a) Famly :When q

4 B.-J. Hong and D.-C. Lu 197 ( ) ( ) ( 1) (1 ) ( ) ( ) Cl r h C lh pe pleh Er lh f Br lh pe e Cp Ar e Epr Dr ef C. (7b) Famly :When r Cl g 1 El fg pe ( p Al ) e qf ( pq Dl ) ef ( q Bl ) eg. (7c) Famly :When l r h C Cpe (Er Cq) f ( Cp Ar ) e ( Cq Eqr Br ) eg (Cpq Epr Dr ) ef. (7d) Substtutng ()()(6)(7) along wth (7a)-(7d) nto Eq.() separately yelds four famles of polynomal equatons for e f g h.settng the coeffcents of e e f e g e h e fg e fh e gh ( 1 ) n to zero yelds a set of over-determned dfferental equatons(odes) p q r l a c ( 1) A B C D E and ( xt ) solvng the ODEs by athematca and Wu elmnaton we can obtan many exact solutons of Eq.(1) accrodng to ()()()()(6). If we let c c1 c c c 1 p 1 q r l a A a1 D a B a E a C we have ( ) F( ) our method contan the mproved general mappng deformaton method[16]etc. Remark 1. Our method proposed here s more general than the extended Rccat equaton ratonal expanson method[6] the extended auxlary functon method [8] the generalzed F-expanson method[1] the generalzed Jacob ellptc functons expanson method[11] and many other algebra expanson methods[911] [ ] etc. Remark. Eq.() and Eq.() can be extended to the followng forms n 1 n n n n ' ( ) a ( t) ( ) u( ) A ( t) ( ) A ( t) ( ) B ( t) ( ) '( ). Where n s usually a postve nteger. If n s a fracton or a negatve nteger we make the followng transformaton: (a) when n d / c s a fracton we let u d/ c ( ) v ( ) then return to determne the balance constant n agan; n (b) when n s a negatve nteger we suppose u( ) v ( ) then return to determne

5 198 New Exact Analytcal Solutons for the General Kdv Equaton the balance constant n agan. Remark. Notced that F( ) ( ) F ( ) ( A B C D E) ( a a a a a ) ( A B C D E) ( ) F ( ) ( a a a a a ) ( A B C D E) 1 We fnd a meanful concluson that ths general method mply a BT of Eq.(1) wth the compatble conons ()()(6)(7) and (7a)-(7d). In the followng we wll use ths method to solve the GVKdV equaton. EXACT SOLUTIONS TO THE GVKDV EQUATION We consder the followng GVKdV equaton [16-]. u ( t) u [ ( t) ( t) x] u ( t) uu ( t) u (8) t x x xxx Where () t () t and () t are arbtrary functons of t. Equaton (8) can be reduced to other more physcal forms [1-6] whch has been dscussed n Ref. [16]. By balancng the hghest-order lnear term u xxx and the nonlnear uu x n (8) we obtan n thus we assume that (8) have the followng solutons: u( ) A ( t) A ( t) ( ) A ( t) ( ) A ( t) ( ) A ( t) ( ) (9) 1 1 ( x t) k( t) x ( t). (1) Where kt () () t A ( t)( 1) are functons of t to be determned later. Substtutng () (1) and (1) nto (8) and settng the coeffcents of ( )( 1 ) and s j x ( ) a ( t) ( )( s 1; j 11 ) to zero yeld a set of over-determned equatons (ODEs) for At () An () t k( t) ( t) and a () t. After solvng the ODEs by athematca we could determne the followng solutons:

6 B.-J. Hong and D.-C. Lu 199 Famly 1 a ' a aa 1 a a (11) A A ( 1) k ' k A a k / A a k / A A 1 8aa 1 a ' k[ A k ( )]. a a Famly a a aa 1 a a (1) 1 A A ( 1) k ' k ' A A A a k / A a k / 1 1 8aa a1 ' k[ A k ( )]. a a 1 Substtutng ()()(6)(7) along wth (7a)-(7d) and (11) nto Eq.() separately yelds an ODEs after solvng the ODEs by athematca and Wu elmnaton we can obtan the followng solutons of Eq.() and Eq.(8) accordng to ()()(6) and (1). (1) (1) Case 1 A B m C m D E F sn m a a m m a m m m m m m 1 1 (1 ) (1 ) (1 6 (1 ) ) a 1(1 m) m 6m m 1 a 8 m( m 1) (1 m)(1 m 6 m) m p 1 q (1 m) m r m l c c c c c 1 sn 1 1( 1) 1 (1 m) msn1msn 1 1 k e x k e [ ( t) ( k k (6(1 m) m 6m m 1)) e ( t)] ( t ) ( t ) ( t ) 1 u k e 1 ( t) (1 m) m (1 6m m (1 m) m) k e sn (1 (1 m) msn msn ) ( m( m 1) 8(1 m)(1 m 6 m) m) k e sn (1 (1 m) msn msn ) ( t) ( t)

7 New Exact Analytcal Solutons for the General Kdv Equaton Case A 1 B m 1 C m D E m 1 F sn a 1 a 1 m a 8 m a 8 1 m a m 1 p q m r l c c c c c sn ( ) 1 m sn cn dn k e x k e [ ( t) ( k k ( 16 m )) e ( t)] u ( t ) ( t ) ( t ) ( t) ( t) ( t) 16 1m k e sn 16(1 m ) k e sn ke ( 1 m sn cn dn ) ( 1 m sn cn dn ) Case A m B m C m D E m F cn a 1 a a 8 m a 8m 8 a m 1 p q 1 r l 1 c c c c c 1 cn ( ) cn sn dn 1 k e x k e [ ( t) ( k k ( m )) e ( t)] u ( t ) ( t ) ( t ) ( t) ( t) ( t) 16( m 1) k e cn 16(1 m ) ke cn ke ( cn sn dn ) ( cn sn dn ) Case A m B m C m D E m F cn a (1 c )[1 ( c 1) m ] a ( c c m c m ) a m 6c m 1 1 a c m a m p 1 q r l c c c c 1 1 ( ) c cn k e x k e [ ( t) (k k (m c m 1))) e ( t)] ( t ) ( t ) ( t ) 8cm k m k ( t) u [ k ( c cn ) ( c cn ) ] e. Remark : u are n full agreement wth the results n Ref.[16]whch contan the results (19) constructed by Zhao n Ref. [17] and u obtaned by Zhu n Ref. [18]...

8 B.-J. Hong and D.-C. Lu 1 Case A m B m C D E m F dn a 1 a m a 8m a 8m 8 m a m m 1 p q m r l 1 c c c c c 1 dn ( ) dn m sn cn 1 k e x k e [ ( t) ( k k (m )) e ( t)] u ( t ) ( t ) ( t ) ( t) ( t) ( t) 16 m(1 m ) k e dn 16 m ( m 1) k e dn ke ( dn m sn cn ) ( dn m sn cn ) Case 6 C1C q C q ( C1 Cq ) C(Cq C1) A B C D C1 E C C C C F ( ) a a C a C q a C C q C C C C q 9 C q ( C C q ) C (C q C ) a sgn[ C C q ] C p q const r 1 l c c1 c c c 1 1 6( 6) C C q ( ) 1 C C ( t) ( t) ( t) 6 ke x ke [ ( t) ( k ( C(C q C1 ) Cq) k ) e ( t)]. ( C C q )( C (C q C ) C q) k u k e ( C1 Cq ) k 1 1 ( t) 6 [ 6( 6) 6 ( 6)]. C Substtutng ()()(6)(7) along wth (7a)-(7d) and (1) nto Eq.() separately yelds an ODEs after solvng the ODEs by athematca and Wu elmnaton we can obtan the followng solutons of Eq.() and Eq.(8) accordng to ()()(6) and (1). Case 7 A B m C m D E m F sn a a q a 6q m 6m 1 a q(1 6m m q ) a ((1 m) q )( m q ) p q m r l 1 c c c c c 1 1

9 New Exact Analytcal Solutons for the General Kdv Equaton sn ( ) qsn7 msn 7 k e x k e [ ( t) (k k (1 6m m q )) e ( t)] ( t ) ( t ) ( t ) 7 8qk k ( t) u7 [ k ( ns 7 q msn 7) ( ns 7 q msn 7) ] e. Case 8 A 1 B m 1 C m D E m 1 F sn a 1 a 1 m a 8 m a 8 1 m a m 1 p q m r l c c c c c sn 8 8( 8) 1m sn 8 cn 8dn 8 k e x k e [ ( t) ( k k ( m )) e ( t)] ( t ) ( t ) ( t ) m k k ( t) u8 [ k ( cs8ds 8 1 m ) ( cs8ds 8 1 m ) ] e. Case 9 A m B m C D E m F dn a 1 a m a 8m a 8m 8 m a m m 1 p q m r l 1 c c c c c 1 dn ( ) dn 9 m sn 9cn 9 1 k e x k e [ ( t) ( k k (m 16)) e ( t)] ( t ) ( t ) ( t ) 9 8mk k ( t) u9 [ k (1 m sd9cd9 ) (1 m sd9cd9 ) ] e. We can gve the numercal smulaton of u and u 7 (see Fgs. 1-).

10 B.-J. Hong and D.-C. Lu (a) (b) Fgure 1. (a) Smulaton of u when k k ( t) ( t) ( t) 1 m.1. (b) Plane graph when t=.

11 New Exact Analytcal Solutons for the General Kdv Equaton (a) (b) Fgure. (a) Smulaton of u 7 when k k ( t) ( t) ( t) 1 m.1. (b) Plane graph when t=.

12 B.-J. Hong and D.-C. Lu Remark : The eght types of explct solutons except u we obtaned here to Eq. (8) are not shown n the prevous lterature to our knowledge. They are new exact solutons of Eq.(8). Solutons u ( 179) are degenerated to soltary-lke solutons when the modulus m 1 and solutons u ( 178) are degenerated to trangular functons solutons when the modulus m. k and k are arbtrary constants n all above cases..conclusion In ths paper we succeed to propose a general algebrac method for fndng new exact solutons of the GVKdV equaton (8). ore mportantly our method s much smple and powerful to fnd new solutons to varous knds of nonlnear evoluton equatons such as KdV equaton Boussnesq equaton zakharov equaton etc. we beleve that ths method should play an mportant role for fndng exact solutons n the mathematcal physcs. Acknowledgments- The authors express ther sncere thanks to the referees for ther careful readng of the manuscrpt and constructve suggeston. The work s supported by the Natonal Nature Scence Foundaton of Chna (Grant No. 6171) the Outstandng Personnel Program n Sx Felds of Jangsu (Grant No. 9188) the Graduate Student Innovaton Project of Jangsu Provnce (Grant No. CXLX1_67) and the Scentfc Research Foundaton of NanJng Insttute of Technology (Grant No. CKJB118).. REFERENCES 1..J. Ablowtz P.A. Clarkson Solton Nonlnear Evoluton Equatons and Inverse Scatterng Cambrdge Unversty Press New York H.Y. Wang X.B. Hu Gegenhas D Toda lattce equaton wth self-consstent sources: Casoratan type solutons blnear Bäcklund transformaton and Lax par Journal of Computatonal and Appled athematcs D.C. Lu B.J. Hong. Bäcklund transformaton and n-solton-lke solutons to the combned KdV-Burgers equaton wth varable coeffcents Internatonal Journal of Nonlnear Scence 1() H.C. Hu X.Y. Tang S.Y. Lou Q.P. LuVarable separaton solutons obtaned from Darboux Transformatons for the asymmetrc Nzhnk-Novkov-Veselov system. Chaos Soltons and Fractals ( ) 7-.

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