Multiple-soliton Solutions for Nonlinear Partial Differential Equations

Transcription

1 Journal of Mathematcs Research; Vol. 7 No. ; ISSN E-ISSN Publshed b Canadan Center of Scence and Educaton Multple-solton Solutons for Nonlnear Partal Dfferental Equatons Yanng Tang & Wean Za Department of Appled Mathematcs Northwestern Poltechncal Unverst X an Chna Correspondence: Yanng Tang Department of Appled Mathematcs Northwestern Poltechncal Unverst X an Chna. E-mal: tanng@nwpu.edu.cn Receved: Ma Accepted: June Onlne Publshed: Jul do:.9/sp.vnp7 URL: Abstract Based on the scale transformaton and the multple ep-functon method the (+)-dmensonal Bot-Leon-Manna-Pempnell (BLMP) equaton and a generalzed Shallow Water Equaton have been solved. The eponental wave solutons whch nclude one-wave two-wave and three-wave solutons have been obtaned. In addton b comparng the solutons obtaned n ths paper wth those solved n the references we fnd that our results are more general. Kewords: Multple ep-functon method BLMP equaton generalzed Shallow Water Equaton multple-solton solutons. Introducton Nonlnear partal dfferental equatons (NLPDEs) have a vtal role n almost all the branches of phscs for eample optcal wavegude fbers flud mechancs and plasma phscs. What s more NLPDEs are ver sgnfcant n appled mathematcs and chemstr. If we need to know the essence of phenomena that have showed b NLPDEs eact solutons of these equatons have to be studed. The stud of solutons of NLPDEs plas a maor part n solton theor and nonlnear scence. The solutons can eplan varous phenomena n man scentfc felds. In recent ears man approaches for solvng eact travelng and soltar wave solutons to NLPDEs have been nvestgated. Among these methods the Panlevéanalss (Hong ) the Darbou transformaton (DT) (Gu & Zhou 99) the nverse scatterng transformaton (Vakhnenko Parkes & Morrson ) the sne cosne method (Yan 99) the Bäcklund transformaton (BT) (Nmmo 98) the etended tanh method (Fan ) the varable separaton approach (Hu Tang Lou & Lu ) the homogeneous balance method (HBM) (Fan 998) the Wronskan technque (Ma ) are effectve and general. However these methods are mostl nvolved travelng wave solutons for NLPDEs. As we all know there are multple wave solutons to NLPDEs for eample multple perodc wave solutons for the Hrota blnear equatons (Ma Zhou & Gao 9) and so on. The multple ep-functon method (Ma Huang & Zhang ) has been proposed and t offers a straghtforward and effcent wa to solve multple wave solutons of NLPDEs (Ma et al. Su Tang & Zhao Ma & Zhu ). In ths paper based on the scale transformaton and ths method the multple solton solutons to the (+)-dmensonal BLMP equaton and a generalzed Shallow Water Equaton are nvestgated. It s worth pontng out that a calculator program reall helps n the process of solvng the ntrcate resultng algebrac equatons. The rest of ths work s planned as follow: at the begnnng a bref eplanaton of the multple ep-functon method s made. In Secton we solve the (+)-dmensonal BLMP equaton b the multple ep-functon method. In Secton we appl the multple ep-functon method on a generalzed Shallow Water Equaton. In the fnal secton we conclude the paper.. Methodolog The authors summarzed the procedure of ths method (Ma et al. ). In a word the processes can be summarzed as: Defnng NLPDEs Transformng NLPDEs and then we solve the algebrac equatons. The emphass of the method s to fnd ratonal solutons n new varables defnng ndvdual waves. We appl the method to solve specfc one-wave two-wave and three-wave solutons for hgh-dmensonal nonlnear equatons. The man steps are shown as follow Ma et al. () gve more detals on steps -. Step. Defnng solvable NLPDEs. 7

2 Journal of Mathematcs Research Vol. 7 No. ; To formulate our soluton steps n the method consder a nonlnear evoluton equaton as f ( z t u u uz ut ). () We ntroduce new varables ( z t) n b solvable NLPDEs for eample the lnear partal dfferental equatons k where k are the angular wave numbers and followng results l m t n. () z are the wave frequences. Solvng Eq. () leads to the k l m z t n () ce where c are arbtrar constants. Each of n descrbes a sngle wave. Step. Transformng NLPDEs We consder ratonal solutons n ( z t) n: where r s u( z t) p( ) q( ) n () n p( n) pr s r s r s n N q( n) qr s r s r s p q are constants to be determned from the Eq.(). We substtute Eq. () nto Eq. () and r s the obtan the soluton M kl m z t kl mz t knlnmnz nt ( ) n k. l m z t kl mz t knln mnznt ( ) n p c e c e c e u( z t) () q c e c e c e Substtutng Eq. () nto Eq. () we obtan the equaton g z t n n ( ). (7) These transformatons make t possble to solve solutons for dfferental equatons drectl b computer algebra sstems. Step. Solvng algebrac sstems Fnall we assume the numerator of g( z t n ) to zero. B solvng these equatons wth the ad of Maple we determne two polnomals p r s q and r s n and then the mult-wave solutons u( z t ) s computed and gven b Eq. (). In the followng sectons we solve the (+)-dmensonal Bot-Leon-Manna-Pempnell equaton and a (+)-dmensonal generalzed Shallow Water equaton b the multple ep-functon method.. Applcaton to the (+)-dmensonal BLMP Equaton The (+)-dmensonal BLMP equaton u u u u u u (8) t () 7

3 Journal of Mathematcs Research Vol. 7 No. ; was derved b Glson Nmmo and Wllo (99). Ma and Fang (9) derved the varable separable solutons of Eq. (8) and nvestgate a lot of new localzed ectatons for eample asmult dromon-soltoffs and fractal-soltons. The (+)-dmensonal BLMP equaton u u u u u ( u u ) u ( u u ) (9) t zt z z z was proposed b Darvsh Naaf Kavtha and Venkatesh () and some eact solutons were obtaned. Wronskan determnant solutons of Eq. (9) were obtaned b Ma and Ba (). In ths paper n order to use the multple ep-functon approach to solve Eq. (9) we frstl make a scale transformaton then the Eq. (9) s reduced equvalentl to u u u u u ( u u ) u ( u u ). () t zt z z z We use the multple ep-functon method to solve the Eq. () and then substtute for n the obtaned results. The new solutons have checked n Eq. (9) and all solutons satsf n the equaton. Net we wll solve one-wave two-wave and three-wave solutons for Eq. (9).. One-wave Solutons Let s assume the lnear condtons to get one-wave soluton as k l z m t () where k l m and are constants. Based on the Eq. () we can obtan the eponental functon solutons We tr the polnomals of degree one k l m z t ce. () p( ) a a q( ) b b. where a a b and b are constants. Substtutng Eq. () and Eq. () nto Eq. () we obtan the equaton () a a c e u( z t). () b k l m zt k l m zt bc e Solvng the above polnomals wth Eq. () we obtan the followng relatons for the algebrac equatons. b ( bk a) a k b () where a b b k and l are constants and a are defned b (). Substtutng Eq. () nto Eq. () we obtan the one-wave solutons of Eq. (). B the scale transformaton we can obtan the one-wave solutons of Eq. (9). We have plotted the one-wave soluton wth a b b k l m z and t n Fgure. 77

4 Journal of Mathematcs Research Vol. 7 No. ; z=t= Fgure. The one-wave soluton wth a b b k l m z and t.. Two-wave Solutons Smlarl we requre the lnear condtons k l m z () t where k l m and are constants. Based on the Eq. () we can obtan the eponental functon solutons We then suppose the polnomals k l m zt ce. (7) p( ) [ k k a ( k k) ] q( ) a (8) where a are constants to be determned. Substtutng Eq. (7) and Eq. (8) nto Eq. () we obtan the equaton [ kc e kce a ( k k) ce ce ] u( z t) ce ce ac e ce k l m z t. B the multple ep-functon method and the Eq. () we can get the followng solutons ( k k)( m l l m ) a ( k k)( m l l m ) k. (9) () where k l and m are constants. Substtutng Eq. () nto Eq. (9) we obtan the two-wave solutons of Eq. (). B the scale transformaton we can obtan the two-wave solutons of Eq. (9). We have plotted the two-wave soluton wth some specal values of ts parameters c c k k l l m m z and t n Fgure. 78

5 Journal of Mathematcs Research Vol. 7 No. ; z=t= Fgure. The two-wave soluton wth c c k k l l m m z and. Three-wave Solutons t. Agan smlarl let s consder the lnear condtons k l m () t z where k l m and are constants. Based on the Eq. () we can obtan the eponental functon solutons In ths condton we set the polnomals as k l m zt ce. () p( ) [ k k k a ( k k) a ( k k) a( k k) aa a( k k k) ] q( ) a a a a aa. () where a a and a the equaton are constants to be determned. Substtutng Eq. () and Eq. () nto Eq. () we obtan u( z t) [ k c e k c e k c e a ( k k ) c e c e a ( k k) ce ce a( k k) ce ce a a a ( k k k ) c e c e c e ] / ( c e ce ce ac e ce ac e ce ace ce aa ace ce c e ) k l m z t. Solvng the above polnomals wth Eq. () we obtan the followng relatons for the algebrac equatons. ( k k )( l l m m ) a ( k k )( l l m m ) k () () 79

6 Journal of Mathematcs Research Vol. 7 No. ; where k l and m are constants. Substtutng Eq. () nto Eq. () we can get the three-wave solutons for Eq. (). B the scale transformaton we solve the solutons to Eq. (9). The determnaton of three-wave solutons confrms the fact that N -wave solutons est for an order. The hgher level wave solutons for N can be obtaned n a parallel manner and do not contan an new free parameters other than a derved for the three-wave solutons. We have plotted the three-wave soluton wth c c c k k k l l l m m m z and t n Fgure. z=t= Fgure. The three-wave soluton wth c c c k k k l l l m m m z and t. It s worth notng that when the two-wave solutons and three-wave solutons were solved b Darvsh et al. () the assumed that the coeffcents of p( ) wthn Eq. (8) and Eq. () are - and obtaned some specal relatons of k l and m. However n ths paper we assume that the coeffcents are and obtan general relaton of k l and m such as Eq. () and Eq. () wth the scale transformaton.e. k l and m are arbtrar constants. So our results are more general. As a specal eample when we set l k m k and k the one-wave two-wave and three-wave solutons obtaned n ths paper correspond to Wronskan determnant solutons for the order N obtaned b Ma and Ba (). The determnaton of three-wave solutons confrms the fact that N -wave solutons est for an order thus the Wronskan determnant solutons for the arbtrar order N obtaned b Ma and Ba () are the specal eamples of the solutons n ths paper.. Applcaton to the (+)-dmensonal Generalzed Shallow Water Equaton The (+)-dmensonal generalzed Shallow Water equaton u u u u u u u () t z was nvestgated b Tan and Gao (99). Solton-tped solutons for Eq. () were obtaned b a generalzed tanh algorthm wth smbolc computaton (Tan & Gao 99). In ths paper n order to use the multple ep-functon method we frstl make a scale transformaton then the Eq. () s reduced equvalentl to u u u u u u u. (7) t z We use the multple ep-functon approach to solve the Eq. () and then substtute for n the obtaned 8

7 Journal of Mathematcs Research Vol. 7 No. ; solutons. The new solutons have checked n Eq. () and all solutons satsf n the equaton. Net we wll solve one-wave two-wave and three-wave solutons for Eq. ().. One-wave Solutons B usng the approach and the Eq. () as well as BLMP equaton we obtan that b ( b k a ) k ( m k l ) a (8) b l where a b b k l and m are constants. Substtutng Eq. (8) nto Eq. () we obtan the one-wave solutons of Eq. (7). B the scale transformaton we can obtan the one-wave solutons of Eq. (). We have plotted the above result for a b b k l m z and t as shown n Fgure. z=t= Fgure. The one-wave soluton wth a b b k l m z andt.. Two-wave Solutons Smlarl b applng the method and Eq. () as well as BLMP equaton we get that a ( k k l l k l k l k k l l l l m k k k l l k l m ml lk mkl ) / (k kl l k l kl kk l l ll mk kk l l kl m ml lk mkl ) k( k ) ( ) l m k k l m. l l (9) where k l and m are constants. Substtutng Eq. (9) nto Eq. (9) we obtan the two-wave solutons of Eq. (7). B the scale transformaton we can obtan the two-wave solutons of Eq. (). We have plotted the above result for c c k k l l m m z and t as shown n Fgure. 8

8 Journal of Mathematcs Research Vol. 7 No. ; z=t= Fgure. The two-wave soluton wth c c k k l l m m z and. Three-wave Solutons t. Agan smlarl b usng the method and Eq. () we obtan that a m ( k k) ( k k) k.. () when l k and k k l l k k l l k l k k l l k k l l k k l l l k a k k l l k k l l k l k k l l k k l l k k l l l k k ( kl ). l () when m k. Substtutng Eq. () and () nto Eq. () we obtan the three-wave solutons of Eq. (7). B the scale transformaton we can obtan the three-wave solutons of Eq. (). The determnaton of three-wave solutons confrms the fact that N -wave solutons est for an order. The hgher level wave solutons for N can be obtaned n a parallel manner and do not contan an new free parameters other than a derved for the three-wave solutons (Wazwaz ). We have plotted the above result obtaned from Eq. () for c c c k k k l l l m m m z andt as shown n Fgure. 8

9 Journal of Mathematcs Research Vol. 7 No. ; z=t= Fgure. The three-wave soluton wth c c c k k k l l l m m m z and t. As a specal eample when we set l k k based on the Eq. () we obtan m k k the one-wave two-wave and three-wave solutons obtaned n ths paper correspond to Wronskan determnant solutons when the order N obtaned b Tang and Su (). The determnaton of three-wave solutons confrms the fact that N -wave solutons est for an order thus Wronskan determnant solutons for the arbtrar order N obtaned b Tang and Su () are the specal eamples of the solutons n ths paper.. Concludng Remarks The multple ep-functon method offers a smple and straghtforward wa to stud eact solutons to NLPDEs. The method has been appled to the (+)-dmensonal BLMP equaton and a generalzed Shallow Water Equaton. The one-wave two-wave and three-wave solutons have been obtaned n ths paper. For some specal values of parameters we obtan some eact solutons. In addton b comparng the solutons obtaned n ths paper wth those solved n the references we fnd that our results are more general. The multple ep-functon method can be appled to obtan the mult-solton solutons of other nonlnear partal dfferental equatons because the multple ep-functon method s effectve n generatng eact solutons of NLPDEs. It s worth pontng out that a few NLPDEs ncludng the (+)-dmensonal YTSF equaton generalzed KP equaton generalzed BKP equaton and BLMP equaton have been solved n (Ma et al. Su et al. Ma & Zhu Darvsh et al. ) when the authors solved the two-wave solutons and three-wave solutons for the frst three equatons the assumed that the coeffcents of p( ) wthn u( z t) were. However the author assumed that the coeffcents were - for BLMP equaton and obtaned some specal relatons among k l and m. In ths paper f for two-wave and three-wave solutons we set that the coeffcents of p( ) wthn u( z t) are we wll meet contradctons n the resultng algebrac sstem. If we suppose the coeffcents of p( ) wthn u( z t) are - we wll not get the more general relatons such as Eq. () and Eq. (). B computaton and analss f we assocate the scale transformaton wth the multple ep method we wll obtan the general solutons for BLMP equaton. An general form of two-waves and three-waves whch does not nvolve an relaton among the angular wave numbers k land m would be more sgnfcant and dvertng. Acknowledgements The work was sponsored b the Seed Foundaton of Innovaton and Creaton for Graduate Students n Northwestern Poltechncal Unverst and the Natonal Natural Scence Foundaton of Chna (Grant No. ). References Darvsh M. T. Naaf M. Kavtha L. & Venkatesh M. (). Star and Step Solton Solutons of the Integrable (+) and (+)-Dmensonal Bot-Leon-Manna-Pempnell Equatons. Commun. Theor. Phs Fan E. G. (998). A note on the homogenous balance method. Phs. Lett. A -. Fan E. G. (). Etended tanh-functon method and ts applcatons to nonlnear equatons. Phs. Lett. A

10 Journal of Mathematcs Research Vol. 7 No. ; Glson C. R. Nmmo J. J. C. & Wllo R. (99). A (+)-dmensonal generalzaton of the AKNS shallow water wave equaton. Phs. Lett. A 8 7- (99). Gu C. H. & Zhou Z. X. (99). On Darbou transformatons for solton equatons n hgh-dmensonal spacetme. Letters n mathematcal phscs Hong L. I. (). Panlevéanalss and Darbou transformaton for a varable-coeffcent Boussnesq sstem n flud dnamcs wth smbolc computaton. Commun. Theor. Phs Hu H. C. Tang X. Y. Lou S. Y. & Lu Q. P. (). Varable separaton solutons obtaned from Darbou transformatons for the asmmetrc Nzhnk Novkov Veselov sstem. Chaos Soltons Fract Ma H. C. & Ba Y. B. (). Wronskan Determnant Solutons for the (+)-Dmensonal Bot-Leon-Manna-Pempnell Equaton. Journal of Appled Mathematcs and Phscs Ma S. H. & Fang J. P. (9). Mult dromon-soltoff and fractal ectatons for (+)-dmensonal Bot Leon Manna Pempnell sstem. Commun. Theor. Phs Ma W. X. (). Wronskans generalzed Wronskans and solutons to the Korteweg de Vres equaton. Chaos Soltons Fract Ma W. X. & Zhu Z. N. (). Solvng the (+)-dmensonal generalzed KP and BKP equatons b the multple ep-functon algorthm. Appl. Math. Comput Ma W. X. Huang T. W. & Zhang Y. (). A multple ep-functon method for nonlnear dfferental equatons and ts applcaton. Phs. Scr Ma W. X. Zhou R. G. & Gao L. (9). Eact one-perodc and two-perodc wave solutons to Hrota blnear equatons n (+) dmensons. Mod. Phs. Lett. A Nmmo J. J. C. (98). A blnear Backlund transformaton for the nonlnear Schrodnger equaton. Phs. Lett. A Su P. P. Tang Y. N. & Zhao Y. (). Multple ep-functon Method for Two Knds of (+)-dmensonal Generalzed BKP Equatons. Chnese Journal of Engneerng Mathematcs Tang Y. N. & Su P. P. (). Two Dfferent Classes of Wronskan Condtons to a (+)-Dmensonal Generalzed Shallow Water Equatons. Internatonal Scholarl Research Notces Tan B. & Gao Y. T. (99). Beond travellng waves: a new algorthm for solvng nonlnear evoluton equatons. Comput. Phs. Commun Vakhnenko V. O. Parkes E. J. & Morrson A. J. (). A Bäcklund transformaton and the nverse scatterng transform method for the generalsed Vakhnenko equaton. Chaos Soltons Fract Wazwaz A. M. (). Multple-Solton Solutons for Etended Shallow Water Wave Equatons. Studes n Mathematcal Scences Yan C. T. (99). A smple transformaton for nonlnear waves. Phs. Lett. A 77 8 (99). Coprghts Coprght for ths artcle s retaned b the author(s) wth frst publcaton rghts granted to the ournal. Ths s an open-access artcle dstrbuted under the terms and condtons of the Creatve Commons Attrbuton lcense ( 8