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Cross-kink multi-sol i ton solutions for th e (3+ 1)-D Jimbo-Miwa equat ion Zhenhui Xu1, Hanlin Chen 2 1Applied Technology College, Southwest University of Science and Technology, Mianyang 621010,PR C hina,e- mail: xuz he nhui@swu s t . e du. cn,xuzhenhui19@16 3.com 2 School of Science, Southwest University of Science and Technology, Mianyang 621010, China. Abstract - In this paper, by using bilinear form and extended three-wave type of ans¨atz approach, we obtain new cross-kink mul ti -soliton solutions of the (3+1)-dimensional J imbo -Miwa equati on, including the periodic b reather-type of kink three-soliton solutions, the cross-kink four-soliton solutions, the doubly periodic breathertype of soliton solutions and the doubly periodic breather-type of cross-kink two-soliton solutions. It is shown that the generalized three-wave method, with the help of symbolic computation, provides an effective and powerful mathematical tool for solving high dimensional nonlinear evolution equations in mathematical physics. Keyword s -Jimbo-Mi wa equation; Extended three-wave method; Cross-kin k mult i -soli ton. 1 Introduction It is well known that many important phenomena in physics and other fields are described by nonlinear partial differential equations. As mathematical models of these phenomena, the investigation of exact solutions is important in mathematical physics. Many methods are availabl e to look for exact solutions of nonlinear evolution equations, such as the inverse scattering method, the Lie group method, the mappi ng method, Exp-function method, ans¨atz technique, three-wave tape of ansatz approach and so on [1-3]. In this paper, we consider the following Jimbo-Miwa equation: 3 3230 xxxy xxy xxyyt xz uuuu uuu+ ++−= (1) which comes from the second member of a KP-hierarchy called Jimbo-Miwa equation firstly introduced by Jimbo-Mi wa [ 4 ]. By means o f the two-solito n method and bilinear methods, the the two-soliton solutions, three-wave solutions of the Jimbo-Miwa were found as well as [5-6]. In this paper, we discuss further the (3 + 1)-dimensional Jimbo-Miwa equation, by using bilinear form and extend thr ee-wave type of ansatz approach, respectively[7-9], Some new cross-kink multi-soliton solutions are obtained. 2 The multi-soliton solutions We assume 2(ln ) x uf= (2) Where (, ,,)ff xyzt= is unknown real function. Substituting Eq.(2) into Eq.(1), we can reduce Eq.(1) into the following Hirota bilinear equation 3 (23)0 x yytxz DDDDDD ff+ −⋅= (3) where the Hirota bilinear operator D is defined by ( ,0mn≥ ) '' '' '' , (,) (,) ()() [(,)(,)] mn xt mn x xt t D D fxtgxt fxtgx t xx tt= = ⋅ ∂∂∂∂ =−− ⋅ ∂∂∂∂ (4) Now we suppose the solution of Eq.(3) as 12 34 cos( )sinh( )cosh( )fe e ξξ δ ηδγδθδ − =++ ++ (5) Where 1111 ax by czdt ξ = +++ , 2222 a xbyczdt η = +++ , 3333 ax byczdt γ = +++ 4444 a xbyczdt θ = +++ and , , ,(1,2,3,4) iii i abcdi= are some constants to be determined later. Substituting Eq.(5) into Eq.(3)and equating all the coefficients of different powers of ,, sin(), cos(), sinh(), cosh(), sinh(), cosh()ee ξξ ηη γγθθ − and consta nt term to zero, we can obtain a set of algebraic equations for , , ,,(1,2,3,4; iii ij abcd i δ = . 1,2,3,4)j= , Solving the system with the aid of Maple, we get the following results: Case (I): Open Journal of Applied Sciences Supplement：2012 world Congress on Engineering and Technology Copyright © 2012 SciRes. 215 This work was supported by Chinese Natural Science Foundation Grant No. 10971169. Sichuan Educationalscience Foundation Grant No.09zc008. 1 324 31 23413 1 22 2 2114141 24 11 1122334 4 0,0, 0, 0, 0,, 0, 0,0, (3 )(3) ,, 22 ,,,. aab b bc cc cdd b a abcacab dd bb δδδδδδδδ == = = = ==== −+ == − == = = (6) Where 241311 2 34 ,,, ,, ,,,a abbc δδδ δ are some free real constant s . Substit uting Eq.(6) into Eq.(5) and ta ki ng 40 δ > , we have 14114 12 1 23134 1 1 2cosh(ln( ))cos() 2 sinh() cosh() fby czaxKt by LzaxHt δ δδ δδ =++ ++ + ++− （7） Where 22 31 2 2114141 1 11 11 1 (3 )(3) ,, . 22 bc a abcacab K LH bb b −+ == = Substituting Eq.(7) into Eq.(2) yields the periodic breather-type of kink three-soliton solutions for Jimbo-Miwa equation as follows: 2121 4341 1 4114121 231341 2[ sin()sinh()] 1 2cosh(ln( ))cos()sinh()cosh() 2 aaxKtaax Ht uby czax Ktby Lzax Ht δδ δδδ δδ +− − = − +++++++− (8) If taking 22 a iA= in Eq.(7), then we ha ve 24114122 231 34 1 1 2cosh(ln( ))cosh() 2 sinh() cosh() fby czAx Kt byLzax Ht δ δδ δδ =++ +− + ++− (9) Where 2 2 211 42 1 ( 3) 0, . 2 A Abc Kb δ + >= Substituting Eq.(9) into Eq.(2) yields the cross-kink four-soliton solutions of Jimbo-Miwa equation as follows: 21224341 2 411412223 1341 2[ sinh()sinh()] 1 2cosh(ln( ))cosh()sinh()cosh() 2 AAxKt aaxHt ubyczAxKt byLzaxHt δδ δδδδ δ −+ − = +++−++ +− (10) FIg(a).The figure of 1 u as 12 11 , ,0 24 t δδ == = Fig(b).The figure of 2 u as 12 11 , ,1 25 t δδ == = If taking 44 a iA= in Eq.(7), then we have 34114 121 231 34 2 1 2cosh(ln( ))cos() 2 sinh() cos() fby czaxKt byLzAx Ht δ δδ δδ =++ ++ +++ + (11) where 2 4 411 42 1 ( 3) 0, . 2 A Abc Hb δ − >= Substituting Eq.(11) into Eq.(2) yields the doubly periodic breather-type of soliton solu t ions for Jimbo-Miwa equation as follows: 21 2 1 4342 3 4114121231342 2[sin()sin()] 1 2cosh(ln( ))cosh()sinh()cos() 2 aaxKt AAxHt uby czax KtbyLzAx Ht δδ δδδδ δ ++ + = − ++++++ +− (12) If taking 4 43 322 ,,aiAbiBiQ δ = == in Eq.(7), then we have 44114121 232 34 2 1 2cosh(ln( ))cos() 2 sin() cos() fby czaxKt QBy LzAxHt δ δδ δ =++ ++ − +++ (13) where 432 ,,ABQ are some free real constants , 31 2 1 Bc Lb = and 3 0. δ > Substituting Eq.(13) into Eq.(2) yields the doubly periodic breather-type of soliton solutions for Jimbo-Miwa equation as follows: 212 1 4342 4 411412123 1342 2[ sin()sin()] 1 2cosh(ln( ))cosh()sin()cos() 2 aaxKt AAxHt uby czaxKtQbyLzAx Ht δδ δ δδδ +++ = − ++++ −+ +− (14) 216 Copyright © 2012 SciRes. 216 Copyright © 2012 SciRes. Case (II): 42 44 1 2312 342 44 44 44 424 4 22 441 44 4 41 3 4123 44 44 44 42 2 422 442 3 1 44141 4 3 44 33 44 422 0,,, 1,0,,,1, 44 3 4(34) 4 ( 2)2 ,, ,, 22 44 ( (3 68)36 ,, 22 aai i a aabb bbc aa aa aac aa a iac c cdd aa aa a aaciaca dd aa δδδ δ − =======−= −− −+− − +− ==== − −− +− + −+− =− =−= 2 222 123 4 4 ) 4() 4(4 )a δδδ + ++ − (15) where 411 2 3 ,,, ,ac δδδ are some free real constants. Substituting Eq.(15) into Eq.(5) and taking 0M> , we have 51 23 1 2cosh(ln())cos( ) 2 sin( )cosh( ) fMM ξ δη δ γδθ = ++ ++ (16) when 0M> . where 4 4 11 4 34 2 a axy czt a ξ − =+++ , 4244 44144 3 44 4(34)4 2 aacaa xz t aa η −+− − = +− , 2 242 444 41 3 44 44 44 2 368 2 2 44 aaa ac xyz t aa aa γ + +− =+++ −− , 2 24 4141 4 43 44 4 44 23 6 2 2 44 acac a ax y zt a aa θ − +− =−++ −− , 4222 222 4231123 4 4 () 4() 4(4 ) a Ma δδδδδδ +− +++ =− . Substituting Eq.(16) into Eq.(2), we obtain the doubly periodic breather-type of cross-kink two-soliton solutions for Jimbo-Miwa equation as follows: 4 14 2 1 43 44 5 1 23 42 1 2[2sinh(ln())sin( )cos()sinh( )] 21 2cosh(ln())cos( )sin( )cosh( ) 2 a aM Ma aa uMM δδ ξηγδθ ξδηδγδθ − ++− − = + +++ (17) Fig(c).The figure of 3 u as 12 11 , ,0 52 t δδ == = Fig(b).The figure of 4 u as 12 1 1,, 0 3t δδ === 3 Conclusion By using bilinear form and extended three-wave type of ans ¨atz approach, we discuss further the (3 +1)-dimensional Jimbo-Miwa equation and find some new cros s -kink multi-soliton solutions. The results show that the extended three-wave tape of ans¨atz approach may provide us with a str aight -forward and effective mathematical tool for seeking multi-wave solutions of higher dimensional inear evolution equations. REFERENCES [1] W.X.Ma, E.G.Fan, Linear superposition principle applying to Hirota bilinear equa tions, Computers and Mathematics with Applications. 61 (2011) 950-959. 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