 Cross-kink multi-sol i ton solutions for th e (3+ 1)-D Jimbo-Miwa equat ion Zhenhui Xu1, Hanlin Chen2 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 2School 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 3230xxxy xxy xxyyt xzuuuu 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 )xuf= (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)0x yytxzDDDDDD ff+ −⋅= (3) where the Hirota bilinear operator D is defined by (,0mn≥) '''''' ,(,) (,)()() [(,)(,)]mnxtmnx xt tD D fxtgxtfxtgx txx tt= =⋅∂∂∂∂=−− ⋅∂∂∂∂ (4) Now we suppose the solution of Eq.(3) as 12 34cos( )sinh( )cosh( )fe eξξδ ηδγδθδ−=++ ++ (5) Where 1111ax by czdtξ= +++,2222a xbyczdtη= +++,3333ax byczdtγ= +++4444a xbyczdtθ= +++ and , , ,(1,2,3,4)iii iabcdi=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()eeξξηη γγθθ− and consta nt term to zero, we can obtain a set of algebraic equations for, , ,,(1,2,3,4;iii ijabcd iδ=.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 32431234131222 211414124111122334 40,0, 0, 0,0,, 0, 0,0,(3 )(3),,22,,,.aab bbccc cddba abcacabddbbδδδδδδδδ== = == ====−+== −== = = (6) Where 241311 2 34,,, ,, ,,,a abbcδδδ δare some free real constant s . Substit uting Eq.(6) into Eq.(5) and ta ki ng40δ>, we have 14114 12 123134 112cosh(ln( ))cos()2sinh() cosh()fby czaxKtby LzaxHtδ δδδδ=++ +++ ++− （7） Where 22312 21141411 1111 1(3 )(3),, .22bca abcacabK LHbb b−+== = Substituting Eq.(7) into Eq.(2) yields the periodic breather-type of kink three-soliton solutions for Jimbo-Miwa equation as follows: 2121 434114114121 2313412[ sin()sinh()]12cosh(ln( ))cos()sinh()cosh()2aaxKtaax Htuby czax Ktby Lzax Htδδδδδ δδ+− −= −+++++++− (8)If taking 22a iA=in Eq.(7), then we ha ve 24114122231 34 112cosh(ln( ))cosh()2sinh() cosh()fby czAx KtbyLzax Htδ δδδδ=++ +−+ ++− (9) Where 22 211421( 3)0, .2A AbcKbδ+>= Substituting Eq.(9) into Eq.(2) yields the cross-kink four-soliton solutions of Jimbo-Miwa equation as follows: 212243412411412223 13412[ sinh()sinh()]12cosh(ln( ))cosh()sinh()cosh()2AAxKt aaxHtubyczAxKt byLzaxHtδδδδδδ δ−+ −=+++−++ +− (10) FIg(a).The figure of 1uas 1211, ,024tδδ== = Fig(b).The figure of 2uas 1211, ,125tδδ== = If taking 44a iA= in Eq.(7), then we have 34114 121231 34 212cosh(ln( ))cos()2sinh() cos()fby czaxKtbyLzAx Htδ δδδδ=++ +++++ + (11) where24 411421( 3)0, .2A AbcHbδ−>= 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 4342341141212313422[sin()sin()]12cosh(ln( ))cosh()sinh()cos()2aaxKt AAxHtuby czax KtbyLzAx Htδδδδδδ δ++ += −++++++ +− (12) If taking 4 43 322,,aiAbiBiQδ= == in Eq.(7), then we have 44114121232 34 212cosh(ln( ))cos()2sin() cos()fby czaxKtQBy LzAxHtδ δδδ=++ ++− +++ (13) where432,,ABQ are some free real constants , 3121BcLb=and30.δ> Substituting Eq.(13) into Eq.(2) yields the doubly periodic breather-type of soliton solutions for Jimbo-Miwa equation as follows: 212 1 43424411412123 13422[ sin()sin()]12cosh(ln( ))cosh()sin()cos()2aaxKt AAxHtuby czaxKtQbyLzAx Htδδδ δδδ+++= −++++ −+ +− (14) 216Copyright © 2012 SciRes.216Copyright © 2012 SciRes. Case (II): 42441 2312 3424444 44424 422 441 444 413 412344444442 2422 442 3 144141 43 4433444220,,, 1,0,,,1,443 4(34) 4( 2)2,, ,,2244((3 68)36,,22aaiia aabb bbcaa aaaac aaa iacc cddaaaaaaaciacaddaaδδδδ−=======−=−−−+− −+−==== −−−+−+ −+−=− =−=2 22212344) 4()4(4 )aδδδ+ ++− (15) where 411 2 3,,, ,acδδδ are some free real constants. Substituting Eq.(15) into Eq.(5) and taking0M>, we have 512312cosh(ln())cos( )2sin( )cosh( )fMMξ δηδ γδθ= ++++ (16) when0M>. where 44114342aaxy cztaξ−=+++,4244441443444(34)42aacaaxz taaη−+− −= +−,2 242444 4134444442 3682244aaa acxyz taaaaγ+ +−=+++−−,2 244141 44344 44423 62244acac aax y ztaaaθ− +−=−++−−,4222 222423112344() 4()4(4 )aMaδδδδδδ+− +++=−. 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: 414 21 434451 234212[2sinh(ln())sin( )cos()sinh( )]212cosh(ln())cos( )sin( )cosh( )2aaM MaaauMMδδξηγδθξδηδγδθ−++− −=+ +++ (17) Fig(c).The figure of 3uas 1211, ,052tδδ== = Fig(b).The figure of 4uas 1211,, 03tδδ=== 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  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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