Explicit and Exact Solutions with Multiple Arbitrary Analytic Functions of Jimbo Miwa Equation

Transcription

1 Avalable at Appl. Appl. Math. ISSN: Vol., Issue (Deceber 009), pp (Prevousl, Vol., No.) Applcatons and Appled Matheatcs: An Internatonal Journal (AAM) Explct and Exact Solutons wth Multple Arbtrar Analtc Functons of Jbo Mwa Equaton Sheng Zhang, Yng-Na Sun, Jn-Me Ba, Lng Dong Departent of Matheatcs Boha Unverst Jnzhou 000, PR Chna Receved: Ma, 009; Accepted: August, 009 Abstract In ths paper, a generalzed F-expanson ethod s used to construct exact solutons of the (+)-densonal Jbo Mwa equaton. As a result, an new and ore general exact solutons are obtaned ncludng sngle and cobned non-degenerate Jacob ellptc functon solutons, hperbolc functon solutons and trgonoetrc functon solutons, each of whch contans sx arbtrar analtc functons. It s shown that wth the ad of sbolc coputaton the generalzed F-expanson ethod a provde a straghtforward and effectve atheatcal tool for solvng nonlnear partal dfferental equatons. Kewords: Nonlnear partal dfferental equatons; F-expanson ethod; Jacob ellptc functon solutons; Hperbolc functon solutons; Trgonoetrc functon solutons MSC (000) No.: 5Q5, 5Q99. Introducton Nonlnear coplex phscal phenoena are related to nonlnear partal dfferental equatons (NLPDEs) whch are nvolved n an felds fro phscs to bolog, chestr, echancs, etc. As atheatcal odels of the phenoena, the nvestgaton of exact solutons of NLPDEs wll help one to understand these phenoena better. Wth the developent of solton theor, an effectve ethods for obtanng exact solutons of NLPDEs have been presented such as Hrota s blnear ethod [Hrota (97)], Bäclund transforaton [Murs (978)], Panlevé expanson [Wess, Tabor and Carnevale (98)], sne-cosne ethod [Yan (996)], hoogeneous balance ethod [Wang (996)], hootop perturbaton ethod 79

2 80 Zhang et al. [El-Shahed (005), He (005), Mohud-Dn and Noor (008)], varatonal teraton ethod [He (999), Noor and Mohud-Dn (008a,b)], tanh-functon ethod [Zaed et al. (00), Abudsala (005), Zhang (007a), Zhang and Xa (006c)], algebrac ethod [Hu (005), Yoba (006), Zhang and Xa (006b)], auxlar equaton ethod [Srendaorej (00), Zhang and Xa (007a,b)], and Exp-functon ethod [Boz and Ber (008), He and Xu (006), Yusufoglu (008), Zhang (008), Zhu (007)]. Recentl, a useful ethod called F-expanson ethod [Wang, et al. (00), Wang and Zhou (00), Zhou et al. (00)] was proposed to construct perodc wave solutons of NLPDEs, whch can be thought of as an over-all generalzaton of Jacob ellptc functon expanson ethod [Lu et al. (00), Fu, et al. (00), Pares et al. (00)]. The F-expanson ethod was later extended n dfferent anners [Chen et al. (005), Lu and Yang (00), Ren and Zhang (006), Wang and L (005), Zhang (007c)]. More recentl, we proposed a generalzed F-expanson ethod [Zhang (006, 007b), Zhang and Xa (006a)] to construct exact solutons whch contan not onl the results obtaned b the ethods of Chen, et al. (005), Wang and L (005), Zhang (007c) but also a seres of new and ore general exact solutons, n whch the restrcton on as erel a lnear functon of tx x x and the restrcton on the coeffcents beng constants are all reoved. In the present paper, we further prove and develop our wor ade n [Zhang (006), (007b), Zhang and Xa (006a)] for obtanng new and ore general exact solutons of the (+)-densonal Jbo Mwa (JM) equaton [Senthlvelan (00)]: u uu u u u u u u () xxx ( ) x xx x x t zz 0 the Panlevé propert, Bäclund transforaton, solton solutons and doubl perodc solutons were nvestgated b Lou (996), Hong and Oh (000), Zhang and Wu (00). The rest of ths paper s organzed as follows. In Secton, we gve the descrpton of the generalzed F-expanson ethod. In Secton, we use ths ethod to obtan ore general exact solutons of the (+)-densonal JM equaton. In Secton, soe conclusons are gven.. Descrpton of the Generalzed F-expanson Method For a gven NLPDE wth ndependent varables x ( tx xx ) and dependent varable u Fuu ( u u u u u u uu u u ) 0 () t x x x xt xt xt tt xx xx xx we see ts solutons n the ore general for n { ( ) ( ) ( ) ( ) ( ) ( )}, u a0 af bf cf F d F F () where a a ( x), ( ), 0 0 a a x b b( x), c c( x), d d( x) ( n) and ( x) are undeterned dfferentable functons, F( ) n () satsfes

3 AAM: Intern. J., Vol., Issue (Deceber 009) [Prevousl, Vol., No. ] 8 ( ) PF ( ) QF ( ) R () F and, hence, holds for F( ) and F ( ) F( ) PF ( ) QF( ) F ( ) (6 PF ( ) Q) F ( ) () 5 F ( ) P F ( ) 0 PQF ( ) ( Q PR) F( ) (5) where P, Q and R are all paraeters, the pre denotes d / d. Gven dfferent values of P, Q and R, the dfferent Jacob ellptc functon solutons F( ) can be obtaned fro (). Soe specal solutons of () are lsted n Table ncludng several new ones whch weren t reported b Zhang (006, 007b), Zhang and Xa (006a). To deterne u explctl, we tae the followng four steps: Step. Deterne the nteger n b balancng the hghest order nonlnear ter(s) and the hghest order partal dervatve of u n (). Step. Substtute () along wth () and (5) nto () and collect all the coeffcents of F ( ) F j ( ) ( 0 j 0 ), then set each coeffcent to zero to derve a set of over-deterned partal dfferental equatons for a,, 0 a b, c, d ( n) and. Step. Solve the sste of over-deterned partal dfferental equatons obtaned n Step for a,, 0 a b, c, d and b use of Matheatca. Step. Select the approprate P, Q, R and F( ) fro Table and substtute the along wth a,, 0 a b, c, d and nto () to obtan non-degenerate Jacob ellptc functon solutons of () (see Table for F ( ) ), fro whch hperbolc functon solutons and trgonoetrc functon solutons can be obtaned n the lt cases when and 0 (see Tables and ). Table. Relatons between values of P,Q, R and correspondng F( ) of () P Q R F( ) ( ) sn, ( ) cn dn cn cd dn ns sn, nc cn nd dn dn dc cn

4 8 Zhang et al. ( ) ( ) sn sc cn sn sd dn cn cs sn dn ( ) ds sn cn ns cs, sn dn nc sc ns ds dn sn cn ( ) cn dn sn sn dn, cn dn cn sn dn dn sn cn sn sn cn dn cn dn Table. Dervatves of Jacob Ellptc Functons sn cndn cd ( )sdnd cn sndn dn sncn ns csds dc ( )ncsc nc scdc nd cdsd sc dcnc cs nsds ds csns sd ndcd

5 AAM: Intern. J., Vol., Issue (Deceber 009) [Prevousl, Vol., No. ] 8 Table. Jacob Ellptc Functons Degenerate nto Hperbolc Functons when sn tanh cn sech dn sech sc snh sd snh cd ns coth nc cosh nd cosh cs csch ds csch dc Table. Jacob Ellptc Functons Degenerate nto Trgonoetrc Functons when 0 sn sn cn cos dn sc tan sd sn cd cos ns csc nc sec nd cs cot ds csc dc sec Rear. The soluton hpothess (), based on the ost exstng wor, s a tral soluton of NLPDEs. It can be easl found that () s ore general than those ntroduced n lterature. To be ore precse, f b c d 0, a0 and a are constants, and s erel a lnear functon of x and t, then () becoes those constructed b Wang, et al. (00), Wang and Zhou (00), Zhou et al. (00). If c d 0, a, 0 a and b are constants, and s erel a lnear functon of x and t, then () reduces to that used b Wang and L (005). If c d 0, then () changes nto the one eploed b Chen et al. (005). If d 0, then () gves the general one proposed b Zhang (007c).. Exact Solutons of the JM Equaton In order to obtan exact solutons of (), we suppose u vx and set the ntegral constant as zero, then () becoes u 6u u uv u vu u 0 (6) u xxx x xx xx t zz v 0. (7) x Accordng to Step, we get n for u and v. To search for explct and exact solutons, we assue that (6) and (7) have the followng foral solutons: u a a F( ) a F ( ) bf ( ) b F ( ) c F( ) c F( ) F( ) 0 df( ) F ( ) df( ) F ( ) (8) v A AF( ) A F ( ) BF ( ) B F ( ) C F( ) C F( ) F( ) 0 DF( ) F ( ) DF( ) F ( ) (9)

6 8 Zhang et al. where a a ( z t), ( ), 0 0 a a z t b b( z t), c c ( z t), d d ( z t), A A ( z t), ( ), 0 0 A A z t B B( z t), C C( z t), D D( z t) ( ), ( zt ), x, s a nonzero constant. Wth the ad of Matheatca, substtutng (8) and (9) along wth () and (5) nto (6) and (7), the left-hand sdes of (6) and (7) are converted nto two polnoals of F ( ) F j ( ) ( 0 j0 ), then settng each coeffcent to zero, we get a set of over-deterned partal dfferental equatons for a, a, a, b, b, c, c, d, d, 0 A, A, A, B, B, C, C, D, 0 D and. Solvng the set of over-deterned partal dfferental equatons b use of Matheatca, we get the followng results: Case. a zg () t g (), t a 0, 0, c P c 0, d 0, a P, b 0, d R, b R, z ( 6 PR Q) ( zg() t g()) t t A0, A 0, A P, B 0, B R, C P, C 0, D 0, D R, l z ztz t zf zf t f f t ( 6 ) ( ) () ( ) (), where f ( ), f (), t f ( ), f (), t g () t and g () t are arbtrar analtc functons of the ndcated varables, l s an arbtrar constant. Case. a zg () t g (), t a 0, 0 a P, b 0, b 0, c P, z Q zg ( ( t) g( t)) 0 t c 0, d 0, d 0, A, A 0, A P, B 0, B 0, C P, C 0, D 0, D 0, l( z 6ztz t) zf ( ) zf () t f ( ) f (), t where f ( ), f (), t f ( ), f (), t g () t and g () t are arbtrar analtc functons of the ndcated varables, l s an arbtrar constant. Case. a zg () t g (), t a 0, 0 a P, b 0, b R, c 0, z Q zg ( () t g()) t t 0 c 0, d 0, d 0, A, A 0, A P, B 0, B R, C 0, C 0, D 0, D 0, l( z 6ztz t) zf( ) zf() t f( ) f(), t

7 AAM: Intern. J., Vol., Issue (Deceber 009) [Prevousl, Vol., No. ] 85 where f ( ), f (), t f ( ), f (), t g () t and g () t are arbtrar analtc functons of the ndcated varables, l s an arbtrar constant. Case. a zg () t g (), t a 0, 0 b 0, b 0, c 0, a P, z Q zg ( () t g()) t 0 t c 0, d 0, d 0, A, A 0, A P, B 0, B 0, C 0, C 0, D 0, D 0, l( z 6ztz t) zf ( ) zf () t f ( ) f (), t where f ( ), f (), t f ( ), f (), t g () t and g () t are arbtrar analtc functons of the ndcated varables, l s an arbtrar constant. Selectng F( ) ns, P, Q( ), R fro Table, usng Case and Table, we obtan cobned non-degenerate Jacob ellptc functon solutons: u zg t g t (0) ( ) ( ) ns sn csds cndn, z ( 6 ) ( zg( t) g( t)) t ns v sn csds cndn, () where x, lz ( 6ztz t) zf( ) zf() t f( ) f(). t Selectng agan F( ) ns cs, P /, Q( ) /, R / fro Table, usng Case and Table, we obtan cobned non-degenerate Jacob ellptc functon solutons: u zg t g t (ns cs ) () () (ns cs ) ds (csds nsds ), () ns cs 6 z ( ) 6 ( zg() t g()) t 6t (ns cs ) v 6 ds (cs ds ns ds ), () (ns cs ) ns cs where x, lz ( 6ztz t) zf( ) zf() t f( ) f(). t

8 86 Zhang et al. In the lt case when, fro () and () we get hperbolc functon solutons: u zg t g t (coth csch ) () () (coth csch ) csch (csch cothcsch ), () coth csch 6 z ( ) 6 ( zg( t) g( t)) 6t (coth csch ) v 6 csch (csch coth csch ), (5) (coth csch ) cothcsch where x, lz ( 6ztz t) zf( ) zf() t f( ) f(). t In the lt case when 0, fro () and () we get trgonoetrc functon solutons: u zg t g t (csc cot ) () () (csc cot ) csc (cotcsc csc ), (6) csc cot 6 z ( ) 6 ( zg( t) g( t)) 6t (csc cot ) v 6 csc (cot csc csc ), (7) (csc cot ) csccot where x, lz ( 6ztz t) zf( ) zf() t f( ) f(). t To the best of our nowledge, the solutons obtaned above have not been reported n the lterature. Fro Cases, we can also obtan other Jacob ellptc functon solutons, hperbolc functon solutons and trgonoetrc functon solutons, here we ot the for splct.. Conclusons In ths paper, we have successfull constructed new and ore general exact solutons wth sx arbtrar analtc functons of the (+)-densonal JM equaton ncludng sngle and cobned non-degenerate Jacob ellptc functon solutons, hperbolc functon solutons and trgonoetrc functon solutons. These solutons contan sx arbtrar analtc functons whch can ae us dscuss the behavors of solutons and also provde us wth enough freedo to construct solutons that a be related to real phscal probles. It a be portant to explan soe phscal phenoena.

9 AAM: Intern. J., Vol., Issue (Deceber 009) [Prevousl, Vol., No. ] 87 Acnowledgents The authors would le to than the referees for ther valuable suggestons and coents. Ths wor was supported b the Natural Scence Foundaton of Educatonal Cottee of Laonng Provnce of Chna under Grant No REFERENCES Abudsala, H. A. (005). On an proved coplex tanh-functon ethod. Int. J. Nonlnear Sc. Nuer. Sul. 6, Boz, A. and Ber, A. (008). Applcaton of Exp-functon ethod for (+)-densonal nonlnear evoluton equatons. Coput. Math. Appl. 56, Chen, J., He, H. S. and Yang, K. Q. (005). A generalzed F-expanson ethod and ts applcaton n hgh-densonal nonlnear evoluton equaton. Coun. Theor. Phs. (Bejng, Chna), El-Shahed, M. (005). Applcaton of He s hootop perturbaton ethod to Volterra s ntegro-dfferental equaton. Int. J. Nonlnear Sc. Nuer. Sul. 6, Fu, Z. T., Lu, S. K., Lu, S. D. and Zhao, Q. (00). New Jacob ellptc functon expanson and new perodc solutons of nonlnear wave equatons. Phs. Lett. A 90, He, J. H. (999). Varatonal teraton ethod a nd of nonlnear analtcal technque: soe exaples. Int. J. Nonlnear Mech., He, J. H. (005). Hootop perturbaton ethod for bfurcaton of nonlnear probles. Int. J. Nonlnear Sc. Nuer. Sul. 6, He, J. H. and Wu, X. H (006). Exp-functon ethod for nonlnear wave equatons. Chaos, Soltons & Fractals 0, Hrota, R. (97). Exact soluton of the Korteweg de Vres equaton for ultple collsons of soltons. Phs. Rev. Lett. 7, 9 9. Hong, W. and Oh, K. S. (000). New soltonc solutons to a (+)-densonal Jbo Mwa equaton. Coput. Math. Appl. 9, 9. Hu, J. Q. (005). An algebrac ethod exactl solvng two hgh-densonal nonlnear evoluton equatons. Chaos, Soltons & Fractals, Lu, J. B. and Yang, K. Q. (00). The extended F-expanson ethod and exact solutons of nonlnear PDEs. Chaos, Soltons & Fractals,. Lu, S. K., Fu, Z. T., Lu, S. D. and Zhao, Q. (00). Jacob ellptc functon expanson ethod and perodc wave solutons of nonlnear wave equatons. Phs. Lett. A 89, Lou, S. Y. (996). Droon-le structures n a (+)-densonal KdV-tpe equaton, J. Phs. A 9, Murs, M.R. (978). Bäclund Transforaton, Sprnger, Berln. Mohud-Dn, S. T. and Noor, M. A. (008). Hootop perturbaton ethod and padé approxants for solvng Flerl Petvashvl equaton. Appl. Appl. Math.: An Int. J.,. Noor, M. A. and Mohud-Dn, S. T. (008a). A relable approach for hgher-order ntegro-dfferental equatons. Appl. Appl. Math.: An Int. J.,

10 88 Zhang et al. Noor, M. A. and Mohud-Dn, S. T. (008b). Solvng hgher densonal ntal boundar value probles b varatonal teraton decoposton ethod. Appl. Appl. Math.: An Int. J., Pares, E. J., Duff, B. R. and Abbott, P. C. (00). The Jacob ellptc-functon ethod for fndng perodc-wave solutons to nonlnear evoluton equatons. Phs. Lett. A 95, Ren, Y. J. and Zhang, H. Q. (006). A generalzed F-expanson ethod to fnd abundant fales of Jacob ellptc functon solutons of the (+)-densonal Nzhn Novov Veselov equaton. Chaos, Soltons & Fractals 7, Senthlvelan, M. (00). On the extended applcatons of hoogeneous balance ethod, Appl. Math. Coput., Srendaorej, Sun, J. (00). Auxlar equaton ethod for solvng nonlnear partal dfferental equatons. Phs. Lett. A 09, Wang, D. S. and Zhang, H. Q. (005). Further proved F-expanson ethod and new exact solutons of Konopelcheno Dubrovs equaton. Chaos, Soltons & Fractals 5, Wang, M. L. (996). Exact soluton for a copound KdV Burgers equatons. Phs. Lett. A, Wang, M. L. and L, X. Z. (005). Applcatons of F-expanson to perodc wave solutons for a new Haltonan apltude equaton. Chaos, Soltons & Fractals, Wang, M. L., Wang, Y. M. and Zhang, J. L. (00). The perodc wave solutons for two sstes of nonlnear wave equatons. Chn. Phs., 8. Wang, M. L. and Zhou, Y. B. (00). The perodc wave solutons for the Klen Gordon Schrödnger equatons. Phs. Lett. A 8, 8 9. Wess, J., Tabor, M. and Carnevale, G. (98). The Panlevé propert for partal dfferental equatons. J. Math. Phs., Yan, C. T. (996). A sple transforaton for nonlnear waves, Phs. Lett. A, Yoba, E. (006). The odfed extended Fan sub-equaton ethod and ts applcaton to the (+)-densonal Broer Kaup Kupershdt equaton. Chaos, Soltons & Fractals 7, Yusufoglu, E. (008). New soltar solutons for the MBBM equatons usng Exp-functon ethod. Phs. Lett. A 7, 6. Zaed, E. M. E., Zedan, H. A. and Gepreel, K. A. (00). Group analss and odfed extended tanh-functon to fnd the nvarant solutons and solton solutons for nonlnear Euler equatons. Int. J. Nonlnear Sc. Nuer. Sul. 5,. Zhang, J. F. and Wu, F. M. (00). Bäclund transforaton and ultple solton solutons for the (+)-densonal Jbo Mwa equaton. Chn. Phs., 5 8. Zhang, S. (006). New exact solutons of the KdV Burgers Kuraoto equaton. Phs. Lett. A 58, 0. Zhang, S. (007a). Sbolc coputaton and new fales of exact non-travellng wave solutons of (+)-densonal Konopelcheno Dubrovs equatons. Chaos, Soltons & Fractals, Zhang, S. (007b). A generalzed F-expanson ethod wth sbolc coputaton exactl solvng Broer-Kaup equatons. Appl. Math. Coput. 89, Zhang, S. (007c). Further proved F-expanson ethod and new exact solutons of Kadostev Petvashvl equaton. Chaos, Soltons & Fractals, Zhang, S. (008). Exp-functon ethod exactl solvng a KdV equaton wth forcng ter. Appl. Math. Coput. 97, 8. Zhang, S. and Xa, T. C. (006a). A generalzed F-expanson ethod and new exact solutons of Konopelcheno-Dubrovs equatons. Appl. Math. Coput. 8,

11 AAM: Intern. J., Vol., Issue (Deceber 009) [Prevousl, Vol., No. ] 89 Zhang, S. and Xa, T. C. (006b). Further proved extended Fan sub-equaton ethod and new exact solutons of the (+)-densonal Broer Kaup Kupershdt equatons. Appl. Math. Coput. 8, Zhang, S. and Xa, T. C. (006c). Sbolc coputaton and new fales of exact non-travellng wave solutons of (+)-densonal Kadostev Petvashvl equaton. Appl. Math. Coput. 8, 9. Zhang, S. and Xa, T. C. (007a). A generalzed auxlar equaton ethod and ts applcaton to (+)-densonal asetrc Nzhn Novov Vesselov equatons. J. Phs. A: Math. Theor. 0, 7 8. Zhang, S. and Xa T. C. (007b). A generalzed new auxlar equaton ethod and ts applcatons to nonlnear partal dfferental equatons. Phs. Lett. A 6, Zhou, Y. B., Wang, M. L. and Wang, Y. M. (00). Perodc wave solutons to a coupled KdV equatons wth varable coeffcents. Phs. Lett. A 08, 6. Zhu, S. D. (007). Exp-functon ethod for the hbrd-lattce sste. Int. J. Nonlnear Sc. Nuer. Sul. 8, 6 6.