Exact Solutions of the Generalized Benjamin Equation and (3 + 1)- Dimensional Gkp Equation by the Extended Tanh Method

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1 Available at Appl Appl Math ISSN: Vol 7, Issue (Jue 0), pp Applicatios ad Applied Mathematics: A Iteratioal Joural (AAM) Exact Solutios of the Geeralized Bejami Equatio ad (3 + )- Dimesioal Gkp Equatio by the Exteded Tah Method N Taghizadeh, M Mirzazadeh ad SR Moosavi Noori Departmet of Pure Mathematics Uiversity of Guila PO Box 94, Rasht, Ira taghizadeh@guilaacir; mirzazadehs@guilaacir moosavioori@mscguilaacir Abstract Received: May 4, 0; Accepted: April 4, 0 I this paper, the exteded tah method is used to costruct exact solutios of the geeralized Bejami ad (3 + )-dimesioal gkp equatio This method is show to be a efficiet method for obtaiig exact solutios of oliear partial differetial equatios It ca be applied to oitegrable equatios as well as to itegrable oes Keywords: Exteded tah method; Geeralized Bejami equatio; (3+)-dimesioal gkp equatio MSC 00: 47F05, 35G0 Itroductio It is well kow that the oliear partial differetial equatios (NLPDEs) are widely used to describe complex pheomea i various fields of scieces, such as physics, biology, chemistry, etc Therefore, seekig exact solutios of NLPDEs is very importat ad sigificat i the oliear scieces I the past decades, great effort has bee made towards this task ad may powerful methods have bee preseted, such as the homogeeous balace method [Khalafallah (009) ad Wag (995, 996)], the modified simplest equatio method [Kudryashov ad Loguiova (008)], the tah method [Malfliet ad Herema (99, 996)], the exteded tah 75

2 76 N Taghizadeh, M Mirzazadeh ad SR Moosavi Noori method [Taghizadeh et al (0) ad Wazwaz (007, 008)], the Jacobia elliptic fuctio expasio method [Liu et al (00)], the first itegral method [Feg et al (00, 005)], ad so o I recet years, may authors have used the exteded tah method to obtai the exact solutios of partial differetial equatios as the method is deemed a efficiet method for obtaiig exact solutios of NLPDEs The aim of this paper is to fid exact solito solutios of the geeralized Bejami equatio ad (3 + )-dimesioal gkp equatio, usig the exteded tah method The Exteded Tah Method ad Tah Method A PDE Fuu (, x, ut, uxx, uxt, uxxx,) 0, () ca be coverted to a ODE Guu (, ', u'', u''',) 0, () upo usig a wave variable x t Eq () is the itegrated as log as all terms cotai derivatives where the itegratio costats are cosidered zeros The stadard tah method is developed by Malfliet [9 ] where the tah is used as a ew variable, sice all derivatives of a tah are represeted by a tah itself Itroducig a ew idepedet variable Y tah( ), x t, (3) leads to the chage of derivatives: d d d dy ( Y ), d Y ( Y ) d ( Y ) d (4) d dy dy The exteded tah method admits the use of the fiite expasio M M k k k (5) k 0 k k u( ) S( Y ) ay b Y,

3 AAM: Iter J, Vol 7, Issue (Jue 0) 77 where M is a positive iteger, i most cases, that will be determied Expasio (5) reduces to the stadard tah method for bk 0, ( k,, M ) Substitutig (5) ito the ODE () results i a algebraic equatio i powers of Y To determie the parameter M, we usually balace the liear terms of highest order i the resultig equatio with the highest order oliear terms We the collect all coefficiets of powers of Y i the resultig equatio where these coefficiets have to vaish This will give a system of algebraic equatios ivolvig the parameters ak ( k 0,, M), bk ( k,, M ), ad Havig determied these parameters we obtai a aalytic solutio u( x, t ) i a closed form 3 Exact Solutios of the Geeralized Bejami Equatio Let us cosider the geeralized Bejami equatio: u ( u u ) u 0, (6) tt x x xxxx where ad are costats This kid of equatio is oe of the most importat NLPDEs, used i the aalysis of log wave i shallow water [Herema et al (986)] By usig the wave variable u( x, t) U ( ), k( x t), (6) becomes the ODE 4 k U ku U k U '' ( ')' '''' 0 (7) Itegratig Equatio (7), twice ad settig the costat of itegratig to zero, we have k '' 0 4 k U U k U (8) Balacig U '' with U i Equatio (8) gives M ( ) M, the M To get a closed form solutio, M should be a iteger To achieve our goal, we use the Trasformatio U( ) V ( ) (9)

4 78 N Taghizadeh, M Mirzazadeh ad SR Moosavi Noori This trasformatio (9) chages Equatio (8) ito the ODE 3 3 k V V k V VV ( ) [( ) ' ( ) ''] 0 (0) Balacig so VV '' with M 3 M, M 3 V i Equatio (0) gives I this case, the exteded tah method i the form (5) admits the use of the fiite expasio b b U( ) S( Y ) a 0 ay ay Y Y () Substitutig the form () ito Equatio (0) ad usig (4), while collectig the coefficiets of Y we obtai: 6 a k ( )( ) a aa 4 k ( ) aa 3 4 k ( ) a 3 [ a a a a ] k ( )[( ) a 8a 6 a a ] k ( ) aa 3 [ a b a aa ] a ( )[ ( 4) (5 ) ] k a0a aa ab

5 AAM: Iter J, Vol 7, Issue (Jue 0) 79 k ( )[ a a a ] 3 [ a a a a a b aa b] k ( )[ a ( ) a 4 a a () ab 4() a b ] 3 0 k ( )[ a a a b] 3 [ a a a b a a b aa b] k ( )[ a0a ( ) aa (9 4) ab ( ) ab ] 0 k ( )[ a ab a b ] 3 [ a b a b a ab a a b ] ( )( )[ ] a0 k ( )[ aa 0 ab 0 ( ) ab 8( ) ab ] 3 k a b Y : k ( )[ a b ab ] 3 [ a b ab a ab a bb ] ( )[ ( ) (9 4) ( ) ] k a0b bb ab ab Y : k ( )[ b a b ] 3 [ a b a b a b abb ] k ( )[ b ( ) b 4 a b () ab 4() a b ] Y : k ( ) bb 3 [ ab a bb ] b ( )[ ( 4) (5 ) ] k a0b bb ab 4 Y : k ( ) b 3 [ ab b b ] k ( )[( ) b 8b 6 ab ] Y :

6 80 N Taghizadeh, M Mirzazadeh ad SR Moosavi Noori 3bb 4 k ( ) bb 3 6 Y : b k ( )( ) b 3 3 Settig these coefficiets equal to zero, ad solvig the resultig system, by usig Maple, we fid the followig sets of solutios: a 0 k( )( ), a 0, a k( )( ), b 0, b 0, k () a 0 k( )( ), a 0, a 0, b 0, b k( )( ), k (3) a 0 k( )( ), a 0, 4 a 8 k( )( ), b 0, b 8 k( )( ), 4 k (4) Recall that U V For 0, the sets ()-(4) give the solitos solutios k( )( ) u( x, t) { sec h [ ( x t)]}, (5)

7 AAM: Iter J, Vol 7, Issue (Jue 0) 8 k( )( ) u( x, t) { csc h [ ( x t)]}, k( )( ) u3( x, t) { (tah [ ( x t)] coth [ ( )])} x t (6) (7) However for 0, we obtai the travellig wave solutios k( )( ) u4( x, t) { sec [ ( x t)]}, (8) k( )( ) u5( x, t) { csc [ ( x t)]}, (9) k( )( ) u6( x, t) { (ta [ ( x t)] cot [ ( x t)])} (0) 4 The (3 + )-dimesioal gkp Equatio The (3 + )-dimesioal gkp equatio, give by ( u 6 u u u ) 3u 3u 0, () t x xxx x yy zz describes the dyamics of solitos ad oliear waves i plasmas physics ad fluid dyamics [Alagesa et al(997)] By usig the wave trasformatio u( x, y, z, t) U ( ), k( x ly mz t), carries Equatio () ito the ODE (3 3 ) '' 6 l m U U U '' 6 U ( U ') k U '''' 0 () Twice itegratig of Equatio (), settig the costat of itegratig to zero, we will have

8 8 N Taghizadeh, M Mirzazadeh ad SR Moosavi Noori 6 l m U U k U (3 3 ) '' 0 (3) Balacig the U '' with U M ( ) M, i Equatio (3) gives M To get a closed form solutio, M should be a iteger To achieve our goal, we use the Trasformatio U( ) V ( ) (4) This trasformatio (4) will chage Equatio (3) ito the ODE ( )(3 3 ) 6 [( ) ' ( ) ''] 0 (5) 3 l m V V k V VV Balacig the VV '' with M 3 M, M 3 V i Equatio (5) gives I this case, the exteded tah method the form (5) admits the use of the fiite expasio b b U( ) S( Y ) a 0 ay ay Y Y (6) Substitutig the form (6) ito Equatio (5) ad usig (4), collectig the coefficiets of Y we obtai: 6 6a k ( )( ) a 3 5

9 AAM: Iter J, Vol 7, Issue (Jue 0) 83 8aa 4 k ( ) aa 4 ( )(3l 3 m ) a 8 [ a a a a ] 0 k ( )[( ) a 8a 6 a a ] 0 3 ( )(3l 3 m ) aa 8 [ a b aaa ] 6 a ( )[ ( 4) (5 ) ] 3 0 k a0a aa ab ( )(3l 3 m )[ a a a ] 8 [ a a a a a b aa b] ( )[ ( ) 4 ( ) 4( ) ] k a a a0a ab ab ( )(3l 3 m )[ aa ab] 8 [ aa abaab aab] ( )[ ( ) (9 4) ( ) ] k a0a aa ab ab 0 ( )(3l 3 m )[ a ab ab] 8 [ ab ab aab aab] 6 ( )[ ( ) 8( ) ] ( )( )[ ] a0 k aa 0 ab 0 ab ab k a b Y : ( )(3l 3 m )[ ab ab ] 8 [ a b ab a ab abb ] ( )[ ( ) (9 4) ( ) ] k a0b bb ab ab Y :

10 84 N Taghizadeh, M Mirzazadeh ad SR Moosavi Noori ( )(3l 3 m )[ b a b ] 8 [ a b a b a b abb ] k ( )[ b ( ) b 4 a b () ab 4() a b ] 0 3 Y : ( )(3l 3 m ) bb 8 [ ab a bb ] 6 b ( )[ ( 4) (5 ) ] 3 0 k a0b bb ab 4 Y : ( )(3l 3 m ) b 8 [ a b b b ] ( )[( ) 8 6 ] 0 k b b a0b 5 Y : 8bb 4 k ( ) bb 6 Y : 6b k ( )( ) b 3 Settig these coefficiets equal to zero, ad solvig the resultig system, by usig Maple, we fid the followig sets of solutios: a0 (3 l 3 m )( )( ), a 0, (3 3 )( )( ), a l m b 0, b 0, (7) 3m 3l k a0 (3 l 3 m )( )( ), a 0, a 0, b 0, (8)

11 AAM: Iter J, Vol 7, Issue (Jue 0) 85 (3 3 )( )( ), b l m 3m 3l k a0 (3 l 3 m )( )( ), a 0, 4 (3 3 )( )( ), a l m b 0, (9) 48 (3 3 )( )( ), b l m 48 Recall that 3m 3l 4 k U V For 3m 3l 0, the sets (7)-(9) give the solitos solutios u x y z t l m (,,, ) { (3 3 )( )( ) 3m 3l h k u( x, y, z, t) { (3l 3 m )( )( ) sec [ ]}, 3m 3l k csc [ ]}, u x y z t l m 48 3(,,, ) { (3 3 )( )( ) h 3m 3l 4 k ( tah [ ] 3m 3l 4 k coth [ ])}, (30) (3) (3) where k( x ly mz t)

12 86 N Taghizadeh, M Mirzazadeh ad SR Moosavi Noori However, for 3m 3l 0, we obtai the travellig wave solutios 3l 3m u4( x, y, z, t) { (3l 3 m )( )( )sec [ ]}, k 3l 3m u5( x, y, z, t) { (3l 3 m )( )( ) csc [ ]}, k u x y z t l m 48 6(,,, ) { (3 3 )( )( ) 3l 3m 3l 3m ( ta [ ] cot [ ])}, 4 k 4 k (35) where k( x ly mz t) (33) (34) 5 Coclusio The exteded tah method has bee successfully applied here to fid the exact solutios for geeralized Bejami equatio ad the (3+)-dimesioal gkp equatio It is also evidet that the proposed method ca be exteded to solve the problems of oliear partial differetial equatios arisig i the theory of solitos ad other areas Ackowledgemet The authors would like to express sicerely thaks to the referees for their useful ad valuable commets REFERENCES Alagesa, T, Uthayakumar, A ad Porsezia, K (997) Pailev aalysis ad Backlud trasformatio for a three-dimesioal Kadomtsev-Petviashvili equatio, Chaos Solitos Fractals, Vol 8, pp Feg, ZS (00) The first itegral method to study the Burgers-Kortewegde Vries equatio, J Phys A, Vol 35, No, pp Feg, ZS ad Wag, XH (00) The first itegral method to the two dimesioal Burgers- KdV equatio, Phys Lett A, Vol 308, pp 73-78

13 AAM: Iter J, Vol 7, Issue (Jue 0) 87 Feg, ZS ad Che, G (005) Solitary wave solutios of the compoud Burgers-Korteweg-de Vries equatio, Physica A, Vol 35, pp Herema, W, Baerjee, P P, Korpel, A, Assato, G, Va Immerzeele, A ad Meerpole, A (986) Exact solitary wave solutios of oliear evolutio ad wave equatios usig a direct algebraic method, J Phys A Math Ge Vol 9, No 5, pp Khalfallah, M (009) New exact travellig wave solutios of the (3+) dimesioal Kadomtsev-Petviashvili (KP) equatio, Commu Noliear Sci Numer Simul, Vol 4, pp Kudryashov, NA ad Loguiova, NB (008) Exteded simplest equatio method for oliear differetial equatios, Appl Math Comput, Vol 05, pp Liu, SK Fu, Z Liu, SD ad Zhao, Q (00) Jacobi elliptic fuctio method ad periodic wave solutios of oliear wave equatios, Phys Lett A, Vol 89, pp Malfliet, W (99) Solitary wave solutios of oliear wave equatios, Am J Phys, Vol 60, No 7, pp Malfliet, W ad Herema, W (996) The tah method: I Exact solutios of oliear evolutio ad wave equatios, Phys Scripta, Vol 54, pp Malfliet, W ad Herema, W (996) The tah method: II Perturbatio techique for coservative systems, Phys Scripta, Vol 54, pp Taghizadeh, N Mirzazadeh, M ad Moosavi Noori, SR (0) Exact solutios of (+)- dimesioal oliear evolutio equatios by usig the exteded tah method, Aust J Basic & Appl Sci Vol 5, No 6, pp Wag, ML (995) Solitrary wave solutio for variat Boussiesq equatio, Phys Lett A, Vol 99, pp 69-7 Wag, ML (996) Applicatia of homogeeous balace method to exact solutios of oliear equatio i mathematical physics, Phys Lett A, Vol 6, pp Wazwaz, AM (007) The exteded tah method for abudat solitary wave solutios of oliear wave equatios, Appl Math Comput Vol 87, pp 3-4 Wazwaz, AM (008) The exteded tah method for the Zakharov-Kuzetsov (ZK) equatio, the modified ZK equatio, ad its geeralized forms, Commu Noliear Sci Numer Simulat, Vol 3, pp

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