New Exact Solutions of the Kawahara Equation using Generalized F-expansion Method

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1 Journal of Mathematcal Control Scence and Applcatons (JMCSA) Vol. No. 1 (January-June, 16), ISSN : Journal of Mathematcal Control Scence and Applcatons (JMCSA) Vol. 1, No. 1, June 7, pp Internatonal Scence Press New Exact Solutons of the Kawahara Equaton usng Generalzed F-expanson Method Sheng Zhang * & Techeng Xa ** Abstract: In ths paper, a generalzed F-expanson method s used to construct exact solutons of the Kawahara equaton. As a result, many new and more general exact travellng wave solutons are obtaned ncludng sngle and combned non-degenerate Jacob ellptc functon solutons, soltary wave solutons and trgonometrc functon solutons. Compared wth the most exstng F-expanson methods, the proposed method gves new and more general exact solutons. More mportantly, the method wth the help of symbolc computaton provdes a powerful mathematcal tool for solvng a great many nonlnear partal dfferental equatons n mathematcal physcs. Keywords: Generalzed F-expanson method, Jacob ellptc functon solutons, Soltary wave solutons, Trgonometrc functon solutons. 1. INTRODUCTION It s well known that nonlnear complex physcal phenomena are related to nonlnear partal dfferental equatons (NLPDEs) whch are nvolved n many felds from physcs to bology, chemstry, mechancs, etc. As mathematcal models of the phenomena, the nvestgaton of exact solutons of NLPDEs wll help one to understand these phenomena better. In the past several decades, many effectve methods for obtanng exact solutons of NLPDEs have been presented, such as nverse scatterng method [1], Hrota s blnear method [], Bäcklund transformaton [3], Panlevé expanson [], sne-cosne method [5], Adoman Pade approxmaton [6], homogenous balance method [7], homotopy perturbaton method [8 1], varatonal method [11 17], asymptotc methods [18], nonperturbatve methods [19], algebrac method [ ], tanh functon method [3 6] and so on. Recently, F-expanson method [7 9] was proposed to construct perodc wave solutons of NLPDEs, whch can be thought of as an over-all generalzaton of Jacob ellptc functon expanson method [3 3]. F- expanson method was later extended n dfferent manners [33 ]. Very recently, by usng a new and more general ansätz we proposed a generalzed F-expanson method [1] whch generalzed the work made n [7 9,35,38 ] to obtan new and more general exact solutons of NLPDEs. Wth the help of Mathematca the generalzed F-expanson method can be appled to a great many NLPDEs. The present paper s motvated by the desre to extend the generalzed F-expanson method [1] to the Kawahara equaton: u uu u u (1) t x xxx xxxxx whch occurs n the theory of magneto-acoustc waves n a plasmas [] and n the theory of shallow water waves wth surface tenson [3]. Srendaorej [] obtaned some travellng wave solutons of Eq. (1) by usng a drect algebrac method. Wazwaz [5] obtaned soltary wave solutons by means of tanh method. * Department of Mathematcs, Boha Unversty, Jnzhou 11, PR Chna, E-mal: zhshaeng@yahoo.com.cn ** Department of Mathematcs, Shangha Unversty, Shangha, PR Chna. 39

2 19 Journal of Mathematcal Control Scence and Applcatons (JMCSA) The rest of ths paper s organzed as follows: n Secton, we gve the descrpton of the generalzed F- expanson method; n Secton 3, we extend ths method to the Kawahare equaton; n Secton, some conclusons are gven.. DESCRIPTION OF THE GENERALIZED F-EXPANSION METHOD For a gven NLPDE wth ndependent varables x ( tx1 x x m) and dependent varable u: F ( uu u u u u u u u u u u ) () we seek ts solutons n the more general form: t x1 x xm x1t xt xmt tt x1 x1 xx xmxm 1 n { ( ) ( ) ( ) ( ) ( ) ( )} 1 u a a F b F c F F d F F, (3) where a a ( x ), ( ) a a x, b b ( x ), c c ( x ), d d ( x ) ( 1 n ) and ( x) are all functons to be determned later, F( ) and F( ) satsfy ( ) ( ) F PF QF ( ) R () and hence holds for F( ) and F( ) 3 F ( ) PF ( ) QF( ) ( ) (6 ( ) ) ( ) (3) F PF Q F () 5 3 F P F PQF Q PR F ( ) ( ) ( ) ( 1 ) ( ) (5) where P, Q and R are all parameters, the prme denotes dd. Gven dfferent values of P, Q and R, the dfferent Jacob ellptc functon solutons F() can be obtaned from Eq. () (see Appendx A). It can be easly found that the ansätz (3) s more general than those n [7 9,35,38 ], to be more precse, f b c d, a and a are constants, and s merely a lnear functon of t x1 x x m, then the ansätz (3) becomes that ntroduced n [7 9]: u n a F ( ) If c d, a and a are constants, and s merely a lnear functon of t x1 x x m, then the ansätz (3) reduces to that constructed n [38,39]: u n n a F ( )

3 New Exact Solutons of the Kawahara Equaton usng Generalzed If c d, then the ansätz (3) changes nto that used n [35]: n 1 u a { a F ( ) b F ( )} If d, then the ansätz (3) converts nto that proposed n []: { ( ) ( ) 1 ( ) ( )} 1 u a a F b F c F F So we may obtan new and more general exact solutons of NLPDEs by usng the ansätz (3). In order to determne u explctly, we take the followng four steps: Step 1. Determne the nteger n by balancng the hghest order nonlnear term(s) and the hghest order partal dervatve of u n Eq. (). l Step. Substtute (3) along wth () and (5) nto Eq. () and collect coeffcents of F ( ) F j ( ) ( l 1 j 1 ), then set each coeffcent to zero to derve a set of over-determned partal dfferental equatons for a, a, b, c, d ( 1 n ) and. Step 3. Solve the system of over-determned partal dfferental equatons obtaned n Step for a, a, b, c, d and by use of Mathematca. Step. Select P, Q, R and F() from Appendx A and substtute them along wth a, a, b, c, d and nto (3) to obtan Jacob ellptc functon solutons of Eq. () (see Appendx B for F( ) ), from whch hyperbolc functon solutons and trgonometrc functon solutons can be obtaned n the lmt cases when m 1 and m (see Appendx C). Remark 1. In order to determne the explct solutons of the partal dfferental equatons derved n Step, we may choose specal forms of a, a, b, c, d and (As we do n Secton 3). 3. EXACT SOLUTIONS OF THE KAWAHARA EQUATION By balancng uu x and u xxxxx n Eq. (1), we get n =. We suppose that Eq. (1) has the followng formal soluton: u a a F( ) a F ( ) a F ( ) a F ( ) b F ( ) b F ( ) b F ( ) b F ( ) c F( ) c F( ) F( ) c F( ) F ( ) c F( ) F ( ) d F( ) F ( ) d F( ) F ( ) d F( ) F ( ) d F( ) F ( ) (6) 3 3 where a, a, b, c, d ( = 1,, 3, ) and are all functon of x and t to be further determned. Wth the ad of Mathematca, substtutng (6) along wth () and (5) nto Eq. (1), the left-hand sde of Eq. l (1) s converted nto a polynomal of F ( ) F j ( ) ( l 1 j 1 ), then settng each coeffcent to 1

4 19 Journal of Mathematcal Control Scence and Applcatons (JMCSA) zero, we get a set of over-determned partal dfferental equatons for a, a, b, c, d and. However, t s very dffcult for us to solve the set of over-determned partal dfferental equatons. As the calculaton goes on, n order to smplfy the work or make the work feasble, we choose specal forms by settng a a ( t ), ( ) a a t, b b ( t ), c c ( t ), d d ( t ), kx, ( t) and k = constant, then we get the followng results: Case 1 a k 91k Q 56 k PR(k Q 1) 1183 k ( Q 18 PR) 57 a1 57k (7) 1 1 a k P(5k Q 1) a3 a 8k P b1 b k R(5k Q 1) (8) 1 b3 b 8 k R c1 k P(k Q 1) c c3 8k P P (9) 1 c d1 d k R(k Q 1) d3 d 8k R R t c (1) ( ) 197 ( 1 ) 19 ( ( 1 ) ) k Q PR k Q Q PR k PR k Q PR Q (11) where and c are arbtrary constants. The sgn ± means that all possble combnatons of + and can be taken n c 3 and d. If the same sgn s used n c 3 and d, then + must be used n a, d and (11). If dfferent sgns are used n c 3 and d, then must be used n a, d and (11). Furthermore, the same sgn must be used n c 1 and c 3. Hereafter, the sgn ± always stands for ths meanng n the smlar crcumstances. Case 31k 91k Q 1183 k ( Q 18 PR) (5 1) a a a k P k Q 57k (1) 1 a3 a 8k P b1 b b3 b c1 k P(k Q 1) () c c 8k P P c d d d d t c (1) where and c are arbtrary constants ( ) 197 ( 36 ) k Q PR k Q Q PR (15) Case 3 31k 36k Q 1898 k ( Q 18 PR) (5 1) a a a k P k Q 57k (16)

5 New Exact Solutons of the Kawahara Equaton usng Generalzed a3 a 168k P b1 b k R(5k Q 1) b3 b 168k R (17) c c c c d d d d t c (18) where and c are arbtrary constants ( 1 ) 168 ( 36 ) k Q PR k Q Q PR (19) Case 31k 91k Q 1183 k ( Q 18 PR) a a a a a 57k () 1 b1 b k R(5k Q 1) b3 b 8k R c1 c c 3 (1) 1 c d1 d k R(k Q 1) d3 d 8k R R t c () where and c are arbtrary constants ( 1 ) 197 ( 36 ) k Q PR k Q Q PR (3) Case 5 31k 36k Q 1898 k ( Q 18 PR) (5 1) a a a k P k Q 57k () a a 168k P b b b b c c (5) where and c are arbtrary constants. c c d d d d t c (6) ( 3 ) 73 ( 9 ) k Q PR k Q Q PR (7) Case 6 31k 36k Q 1898 k ( Q 18 PR) a a a a a 57k (8) 8 b1 b k R(5k Q 1) b3 b 168k R c1 c (9) 3

6 19 Journal of Mathematcal Control Scence and Applcatons (JMCSA) c c d d d d t c (3) ( 3 ) 73 ( 9 ) k Q PR k Q Q PR (31) where w and c are arbtrary constants. Substtutng Cases 1 6 nto (6) respectvely, we have sx knds of formal solutons of Eq. (1): u k 91k Q 56 k PR(k Q 1) 1183 k ( Q 18 PR) 57 57k 1 1 k P(5k Q 1) F ( ) 8 k P F ( ) k R(5k Q 1) F ( ) 8 k R F ( ) 1 1 k P(k Q 1) F( ) 8 k P PF( ) F ( ) k R(k Q 1) F( ) F ( ) 8 k R RF ( ) F ( ) (3) where kx t c, k s determned by (11). 31k 91k Q 1183 k ( Q 18 PR) 57 1 u k P(5k Q 1) F ( ) 57k 1 8 k P F ( ) k P(k Q 1) F( ) 8 k P PF( ) F ( ) (33) where kx t c, k s determned by (15). 31k 36k Q 1898 k ( Q 18 PR) 57 8 u k P(5k Q 1) F ( ) 57k 8 (3) 168 k P F ( ) k R(5k Q 1) F ( ) 168 k R F ( ) where kx t c, k s determned by (19). 31k 91k Q 1183 k ( Q 18 PR) 57 1 u k R(5k Q 1) F ( ) 57k 1 (35) 8 k R F ( ) k R(k Q 1) F ( ) F ( ) 8 k R RF ( ) F ( )

7 New Exact Solutons of the Kawahara Equaton usng Generalzed where kx t c, k s determned by (3). 31k 36k Q 1898 k ( Q 18 PR) 57 8 u k P(5k Q 1) F ( ) 57k 168 k P F ( ) (36) where kx t c, k s determned by (7). 31k 36k Q 1898 k ( Q 18 PR) 57 8 u k R(5k Q 1) F ( ) 57k 168 k R F ( ) (37) where kx t c, k s determned by (31). From Appendx A, choosng F( ) ns, P 1, Q (1 m ), R m, nsertng them nto (3) and usng Appendx B, we obtan combned non-degenerate Jacob ellptc functon soluton of Eq.(1): u k 91 k (1 m ) 56 k m( k (1 m ) 1) 1183 k ((1 m ) 18 m ) 57 57k 1 1 k (5 k (1 m ) 1) ns 8 k ns k m (5 k (1 m ) 1) sn 1 8 k m sn k ( k (1 m ) 1)csds 8k csds ns 1 k m( k (1 m ) 1)cndn 8k m cndnsn (38) 3 where kx t c, k s determned by (11) wth P 1, Q (1 m ) and R m. In the lmt case when m 1, from (38) we obtan soltary wave soluton of Eq. (1): k 18k 56 k (6k 1) 133k 57 1 u k (1k 1)coth 57k k coth k (1k 1) tanh 8 k tanh k (6k 1) csch 1 8 k cosh coth k (6k 1) sech 8k sech tanh (39) where kx t c, k s determned by (11) wth P = 1, Q = and R = 1. 5

8 196 Journal of Mathematcal Control Scence and Applcatons (JMCSA) When m, from (38) we obtan trgonometrc functon soluton of Eq. (1): k 91k 1183k 57 1 u k (5k 1) csc 8k csc 57k 1 k (k 1)cotcsc 8k cot csc () 3 where kx t c, k s determned n (11) wth P 1, Q 1 and R. Choosng F( ) ns cs, P 1, Q (1 m ), R 1, nsertng them nto (3) and usng Appendx B, we obtan combned non-degenerate Jacob ellptc functon soluton of Eq. (1): u k 18 k (1 m ) 8 k ( k (1 m ) ) 1183 k ((1 m ) 7) 8 8k k (6 k (1 m ) 1)(ns cs ) k (ns cs ) k (6 k (1 m ) 1) k (ns cs ) (ns cs ) 35 k ( k (1 m ) )(csds nsds ) 15 k (csds nsds )(ns cs ) 35 ds ds k ( k (1 m ) ) 15k ns cs (ns cs ) 3 (1) where kx t c, k s determned by (11) wth P 1, Q (1 m ) and 1 In the lmt case when m 1, from (1) we obtan soltary wave soluton of Eq. (1): R k 18k 8 k (k ) 3758k 8 35 u k (6k 1)(coth csch ) 8k k (coth csch ) k (6k 1) k (coth csch ) (coth csch ) 35 k (k )( csch cothcsch ) 15 k ( csch cothcsch )(coth csch ) 35 csch csch ( ) 15 3 coth csch k k k (coth csch ) () 6

9 New Exact Solutons of the Kawahara Equaton usng Generalzed where kx wt c, k s determned by (11) wth P = 1/, Q = 1/ and R = 1/. When m, from (1) we obtan trgonometrc functon soluton of Eq. (1): k 18k 8 k (k ) 3758k 8 35 u k (6k 1)(csc cot ) 8k k (csc cot ) k (6k 1) k (csc cot ) (csc cot ) 35 k (k )(cotcsc csc ) 15 k (cotcsc csc )(csc cot ) 35 csc csc k (k ) 15k csc cot (csc cot ) 3 (3) where kx t c, k s determned by (11) wth P = 1/, Q = 1/ and R = 1/. Wth the ad of Appendces A, B and C, from (3) (37) we can obtan other sngle and combned Jacob ellptc functon solutons, soltary wave solutons and trgonometrc functon solutons of Eq. (1), we omt them here for smplcty. Remark. All solutons obtaned from Cases 1 and can not been obtaned by the F-expanson methods [7 9,33 ]. To the best of our knowledge, they are new and have not been reported n former lterature. Wth the ad of Mathematca, we have verfed all solutons obtaned n ths paper by puttng them back nto the orgnal Eq. (1).. CONCLUSION In ths paper, the generalzed F-expanson method has been successfully used to obtan new and more general exact solutons of the Kawahara equaton. These exact solutons nclude sngle and combned non-degenerate Jacob ellptc functon solutons, soltary wave solutons and trgonometrc functon solutons. To our best knowledge, these solutons have not been reported n former lterature. It may be mportant to explan some physcal phenomena. It s shown that the generalzed F-expanson method wth the help of Mathematca provdes a powerful mathematcal tool for obtanng more general exact solutons of a great many NLPDEs n mathematcal physcs, such as the (3+1)-dmensonal Kadomtsev Petvashvl (KP) equaton, the (+1)-dmensonal Broer Kaup Kupershmdt (BKK) equatons, Nzhnk Novkov Vesselov (NNV) equatons and so on. Compared wth the most exstng F-expanson methods [7 9,35,38 ], the proposed method gves new and more general exact solutons. Its applcatons are worth further studyng. ACKNOWLEDGMENTS Ths work was supported by the Natural Scence Foundaton of Educatonal Commttee of Laonng Provnce of Chna (6). 7

10 198 Journal of Mathematcal Control Scence and Applcatons (JMCSA) APPENDIX A Gven dfferent values of P, Q and R, the dfferent Jacob ellptc functon solutons F( ) of Eq. () can be obtaned, whch are lsted n Table 1. Table 1 Relatons between values of (P, Q, R) and correspondng F( ) n Eq. () P Q R F() m (1+m ) 1 sn, cdx= cn dn m m 1 1 m cn 1 m m 1 dn 1 (1+m ) m ns= (sn) 1, dc= dn cn 1 m m 1 m nc= (cn) 1 m 1 m 1 nd= (dn) 1 1-m m 1 sc= sn cn m (1 m ) m 1 1 sd = sn dn 1 m 1 m cs = cn sn 1 m 1 m (1 m ) ds = dn sn 1 1 m 1 ns ± cs 1 m 1 m 1 m nc ± sc 1 m m ns ± ds m m m sn ± cn Dervatves of Jacob ellptc functons APPENDIX B sn cn dn cd (1 m )sd nd cn sn dn dn m sn cn ns cs ds dc (1 m )nc sc nc sc dc nd m cd sd sc dcnc cs nsds ds csns sd ndcd 8

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13 New Exact Solutons of the Kawahara Equaton usng Generalzed... 1 [1] S. Zhang, New exact solutons of the KdV Burgers Kuramoto equaton, Phys. Lett. A, Vol. 358, pp. 1-, 6. [] T. Kawahara, Oscllatory soltary waves n dspersve meda, J. Phys. Sco. Jpn., Vol. 33, pp. 6-6, 197. [3] J.K. Hunter and J. Scheule, Exstence of perturbed soltary wave solutons to a model equaton for water waves, Physca D, Vol. 3, pp , [] Srendaorej, New exact travellng wave solutons for the Kawahara and modfed Kawahara equatons, Chaos, Soltons & Fractals, Vol. 19, pp ,. [5] A.M. Wazwaz, New soltary wave solutons to the Kuramoto Svashnsky and the Kawahara equatons, Appl. Math. Comput., Vol. 18, pp , 6. 51