EXACT TRAVELLING WAVE SOLUTIONS FOR THREE NONLINEAR EVOLUTION EQUATIONS BY A BERNOULLI SUB-ODE METHOD

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1 10.pdf EXACT TRAVELLING WAVE SOLUTIONS FOR THREE NONLINEAR EVOLUTION EQUATIONS BY A BERNOULLI SUB-ODE METHOD Chengbo Tan & Qnghua Feng * School of Scence, Shandong Unversty of Technology, Zhangzhou Road 1, Zbo, Shandong, Chna, *Eal: fqhua@sna.co ABSTRACT In ths paper, based on a proposed Bernoull sub-ode ethod and wth the ad of atheatcal software Maple, we derve soe new exact travellng wave solutons for the (3+1) densonal Jbo-Mva equaton, SRLW equaton and generalzed Burgers equaton. These solutons are of new fors so far n the lterature. Keywords: Bernoull sub-ode ethod, Travellng wave solutons, Exact soluton, Evoluton equaton, (3+1) densonal Jbo-Mva equaton, SRLW equaton, Generalzed Burgers equaton 1. INTRODUCTION The nonlnear phenoena exst n all the felds ncludng ether the scentfc work or engneerng felds, such as flud echancs, plasa physcs, optcal fbers, bology, sold state physcs, checal kneatcs, checal physcs and so on. It s well known that any nonlnear evoluton equatons (NLEEs) are wdely used to descrbe these coplex phenoena. Research on solutons of NLEEs s popular. So, the powerful and effcent ethods to fnd analytc solutons and nuercal solutons of nonlnear equatons have drawn a lot of nterest by a dverse group of scentsts. Many effcent ethods have been presented so far. Durng the past few decades searchng for explct solutons of nonlnear evoluton equatons by usng varous dfferent ethods have been the an goal for any researchers, and any powerful ethods for constructng exact solutons of nonlnear evoluton equatons have been establshed and developed such as the nverse scatterng transfor, the Darboux transfor, the tanh-functon expanson and ts varous extenson, the Jacob ellptc functon expanson, the hoogeneous balance ethod, the sne-cosne ethod, the rank analyss ethod, the exp-functon expanson ethod and so on [1-13]. In ths paper, we propose a Bernoull sub-ode ethod to construct exact travellng wave solutons for NLEES. The rest of the paper s organzed as follows. In Secton, we descrbe the Bernoull sub-ode ethod for fndng travellng wave solutons of nonlnear evoluton equatons, and gve the an steps of the ethod. In the subsequent sectons, we wll apply the Bernoull sub-ode ethod to fnd exact travellng wave solutons for the (3+1) densonal Jbo-Mva equaton, SRLW equaton, and generalzed Burgers equaton. In the last secton, soe conclusons are presented.. DESCRIPTION OF THE BERNOULLI SUB-ODE METHOD In ths secton we present the solutons of the followng ODE: G' G G (1) 0, G G( ). When 0, Eq. (1) s the type of Bernoull equaton, and we can obtan the soluton as 1 () G de d s an arbtrary constant. Suppose that a nonlnear equaton, say n two or three ndependent varables xy, and t, s gven by P( u, ut, ux, uy, utt, uxt, uyt, uxx, uyy...) 0 (3) u u( x, y, t) s an unknown functon, P s a polynoal n u u( x, y, t) and ts varous partal dervatves, n whch the hghest order dervatves and nonlnear ters are nvolved. Step 1.We suppose that u( x, y, t) u( ), ( x, y, t) (4) The travellng wave varable (4) perts us reducng Eq. (3) to an ODE for u u( ) P( u, u', u'',...) 0 (5) 35

2 Step. Suppose that the soluton of (5) can be expressed by a polynoal n G as follows: 1 u( ) G G... (6) 1,... are constants to be deterned later, 0. The postve G G( ) satsfes Eq. (1), and 1 nteger can be deterned by consderng the hoogeneous balance between the hghest order dervatves and non-lnear ters appearng n (5). Step 3. Substtutng (6) nto (5) and usng (1), collectng all ters wth the sae order of G together, the left-hand sde of Eq. (5) s converted nto another polynoal n G. Equatng each coeffcent of ths polynoal to zero, yelds a set of algebrac equatons for, 1,...,. Step 4. Solvng the algebrac equatons syste n Step 3, and by usng the solutons of Eq. (1), we can construct the travellng wave solutons of the nonlnear evoluton equaton (5). In the subsequent sectons we wll llustrate the valdty of the proposed ethod by applyng t to solve several nonlnear evoluton equatons. 3. APPLICATION OF BERNOULLI SUB-ODE METHOD FOR THE (3+1) DIMENSIONAL JIMBO-MIVA EQUATION In ths secton, we consder the followng (3+1) densonal Jbo-Mva equaton: u 3u u 3u u u 3u 0 (7) Suppose that xxxy y xx x xy yt xz u( x, y, z, t) u( ), kx y rz t (8) k,, r, are constants that to be deterned later. By (8), (7) s converted nto an ODE 3 (4) k u k u u kr u 6 ' '' ( 3 ) '' 0 (9) Integratng (9) once we obtan 3 k u ''' 3 k ( u ') ( 3 kr) u' g (10) g s the ntegraton constant to be deterned later. Suppose that the soluton of (10) can be expressed by a polynoal n G as follows: u( ) a G (11) 0 a are constants, and G G( ) satsfes Eq. (1). Balancng the order of rewrtten as u ''' and ( u ') n Eq. (10), we have 3 1.So Eq. (11) can be u( ) a G a, a 0 (1) a1, a 0 are constants to be deterned later. Substtutng (1) nto (10) and collectng all the ters wth the sae power of G together, equatng each coeffcent to zero, yelds a set of sultaneous algebrac equatons as follows: G 0 : g 0 G : a 3kra k a G : 3a kr 7k a 3k a a G : 1k a 6k a G :6k a 3k a Solvng the algebrac equatons above, yelds: 36

3 1 k(3 r k ) a1 k, a0 a0, k k, r r,, g 0, (13) k, r, are arbtrary nonzero constants. Substtutng (13) nto (1), we get that 1 k(3 r k ) u( ) kg a, 0 kx ry z t (14) Cobnng wth Eq. (1.), we can obtan the travellng wave solutons of (7) as follows: 1 u( ) k( ) a de 0 (15) 1 k(3 r k ) kx ry z t, and, are arbtrary nonzero constants. Reark 1. Our result (15) (7). s new exact travellng wave solutons for the (3+1) densonal Jbo-Mva equaton 4. APPLICATION OF BERNOULLI SUB-ODE METHOD FOR SRLW EQUATION In ths secton, we wll consder the followng SRLW equaton: u u u uu u u 0 (16) xxtt tt xx xt x t Suppose that u( x, t) u( ), kx t (17) k are constants that to be deterned later. By (17), (16) s converted nto an ODE, (4) ( k ) u'' kuu '' k( u ') k u 0 (18) Integratng (18) once we obtan ( ) ' ' ''' k u k uu k u g (19) g s the ntegraton constant to be deterned later.. Suppose that the soluton of (19) can be expressed by a polynoal n G as follows: u( ) a G (0) 0 a are constants, and G G( ) satsfes Eq. (1). Balancng the order of u ''' and uu ' n Eq. (19), we have 3 1. So Eq. (0) can be rewrtten as u( ) a G a G a, a 0 (1) 1 0 a, a, a are constants to be deterned later. 1 0 Substtutng (1) nto (19) and collectng all the ters wth the sae power of G together, equatng each coeffcent to zero, yelds a set of sultaneous algebrac equatons as follows: G 0 : g 0 G : ka a a k a k a G : a a 7k a ka a ka ka a 8k a k a k a G : k a 3ka a ka 1k a 38k a a ka a G : 54k a 6k a ka 3kka a

4 G : 4k a ka Solvng the algebrac equatons above, yelds: k k a 1 k, a1 1 k, a0, k k, g 0, () k k, are arbtrary nonzero constants. Substtutng () nto (1), we get that k k u( ) 1k G 1kG, kx t (3) k Cobnng wth Eq. (1.), we can obtan the travellng wave solutons of (16) as follows: 1 1 k k u( ) 1 k ( ) 1 k( ) de de k kx t, and, are arbtrary nonzero constants. (4) Reark. The soluton denoted n (4) s new exact travellng wave solutons for the SRLW equaton (16). 5. APPLICATION OF BERNOULLI SUB-ODE METHOD FOR GENERALIZED BURGERS EQUATION In ths secton, we wll consder the followng generalzed Burgers equaton: u u u u 0 (5) t xx x Suppose that u( x, t) u( ), x ct (6) c s a constant that to be deterned later. By (6), (5) s converted nto an ODE cu ' u'' u u' 0 (7) Suppose that the soluton of (7) can be expressed by a polynoal n G as follows: u( ) a G (8) 0 a are constants, and G G( ) Balancng the order of satsfes Eq. (1). uu ' and u '' n Eq. (7), we have 1 n n 1 n n. So we ake a varable u v 1, and (7) s converted nto cvv ' (1 )( v') vv '' v v' 0 (9) Suppose that the soluton of (9) can be expressed by a polynoal n G as follows: l v( ) bg (30) 0 b are constants, and G G( ) Balancng the order of can be rewrtten as vv ' and satsfes Eq. (1.1). vv '' n Eq. (9), we have l l 1 l l l 1. So Eq. (30) v( ) bg b, b 0 (31) Substtutng (31) nto (9) and collectng all the ters wth the sae power of G together, equatng each coeffcent to zero, yelds a set of sultaneous algebrac equatons as follows: 38

5 G : a a ca a a a G : a a a a ca a a ca 3 a a G : a a ca a a a a a G : a a a Solvng the algebrac equatons above, yelds: Case 1: ( 1) a1, a0 0, c (3) Substtutng (31) nto (30), we get that ( 1) u( ) G, x t (33) Cobnng wth Eq. (1.), we can obtan the travellng wave solutons of (5) as follows: ( 1) 1 u( ) ( ) (34) de, are arbtrary nonzero constants. Snce x t, then furtherore we have ( 1) 1 u( x, y, t) ( ) (35) ( x t) de Case : ( 1) ( 1) a1, a0, c (36) Substtutng (31) nto (30), we get that ( 1) ( 1) u( ) G, x t (37) Cobnng wth Eq. (1.), we can obtan the travellng wave solutons of (5) as follows: ( 1) 1 ( 1) u( ) ( ) (38) de, are arbtrary nonzero constants. Snce x t, then furtherore we have ( 1) 1 ( 1) u( x, y, t) ( ) (39) ( x t) de Reark 3. Our results (35) and (39) are new exact travellng wave solutons for the generalzed Burgers equaton (5). 6. CONCLUSIONS We have seen that soe new travellng wave soluton of the (3+1) densonal Jbo-Mva equaton, SRLW equaton, and generalzed Burgers equaton are successfully found by usng the Bernoull sub-ode ethod. As one can see, ths ethod s concse, powerful, and effectve. Also ths ethod can be generalzed to solve any other nonlnear evoluton equatons. 39

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