 Journal of Applied Mathematics and Physics, 2013, 1, 18-24 Published Online November 2013 (http://www.scirp.org/journal/jamp) http://dx.doi.org/10.4236/jamp.2013.15004 Open Access JAMP Wronskian Determinant Solutions for the (3 + 1)-Dimensional Boiti-Leon-Manna-Pempinelli Equation Hongcai Ma, Yongbin Bai Department of Applied Mathematics, Donghua University, Shanghai, China Email: hongcaima@hotmail.com Received August 4, 2013; revised September 4, 2013; accepted October 1, 2013 Copyright © 2013 Hongcai Ma, Yongbin Bai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In this paper, we consider (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equ ation. Based on the bilinear form, we derive exact solutions of (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli (BLMP) equation b y using the Wronskian technique, which include rational solutions, soliton solutions, positons and negatons. Keywords: (3 + 1)-Dimensional Boiti-Leon-Manna-Pempinelli Equ ation; The Wronskian Technique; Soliton; Negaton; Positon 1. Introduction The Wronskian technique is introduced by Freeman and Nimmo . After that, many researches are based on the Wronskian technique. The (2 + 1)-dimensional BLMP equation was first de- rived in : 33ytxxxyxx yx yuuuu uu 0 (1) where and subscripts represent partial differentiation with respect to the given variable. This equation was used to describe the (2 + 1)-dimensional interaction of the Riemann wave propagated along the y-axis with a long wave propaga ted along the x-ax is. The Painlevé analysis, Lax pairs, Bäcklund transformation, symmetry, similarity reductions and new exact solutions of the (2 + 1)-dimensional BLMP equation are given in [2-4]. In , based on the binary Bell polynomials, the bilinear form for the BLMP equation is obtained. New solutions of (2 + 1)-dimensional BLMP equation from Wronskian formalism and the Hirota method are ob- tained in [6,7]. ,,uuxytThe (3 + 1)-dimensional BLMP equation 330yt ztxxxy xxxzxxy xzxx yzuuuuuuuuu u  (2) which was introduced in  has the bilinear form 33 0yt ztyxzxDDDDDDDDf f  (3) just by substituting 2ln, ,,xufxyz tinto equation (2), where the bilinear differential operator D is defined by Hirota  as 0, 0,,,,mntxmnsymnDDatx btxat sxybt sx ysy 2. Wronskian Formulation Solutions determined by 2lnxu fto the Equation (2) are called Wronskian solutions, where    1201 111 101 122 201 1,,, 1,1NNNNNN NfW NN    , (4) and  20,,1,1jiii ijjix N. (5) H. C. MA, Y. B. BAI 19Lemma 1 ,,,,,,,,,,,, 0DabDcdDacDbdDad Dbc, (6) w matrix, and are n-dimensional columors. -dimensiona real consta bu hen we have here D is 2NNn vect,,,abcdLemma 2 Set to be an nl column vector, and 1, ,jrj n to be ant 1, ,jbj n t not to be zero. T12,,,, ,,NNi jNrbbrba, (7) 12,,Nb bbwhere Lemma 3 The following equalities hold: 11ijT112 2,,,jjjNNjrbrbr brb.  211 111 1Nn Nii iiiiii jNN N   . (8) Proposition. Assuming that ,,,iixyzt,N (where ) has continuous de- 0,, ,,1,2,txyzirivative up to any order and satisfies the following linear differential conditions ,,, ,,,14,,,Nitixxxixxijjiyixizixi,   (9) then 1fN defined by Equation (4) solves the bil- inear Equation (2). Proof. Using the conditions (9), we get that 2, ,zyxfffNN 3,1,2,1,xz xy xxfffNNNNN  4,2, 1,23, 1, 12,2xxz xxyxxxf ffNNNNNNNNN  5,3,,2,, 1,34,2, 1, 133,1,223,,1 2,3xxxz xxxy xxxxfffNN NNNNNNNNNNNNN NN  44,2,1,43,1,142,2tfNNNNNNN NN  ,45,3,2,1,43,,142,3yt zt xtfff NNNNNNNN.N      Hence, we have N3325,3,,2,,1,4,2,1,123,,13, 1,22,3243, 1, 12,63,1,2,1.yt zt yx zxDD DD DDDDffNNNNNNNN NNNNNNNNNNNNNNNNNN  (10) With the help of Lemma 2 and Lemma 3, we obtain 61NN261 5,3,,2,,1,4,2,1,1 23,,13, 1,22,363,1, 2,1.NNNN NNNNNN NNNNNNNNNNNNN    (11) Substituting Equation (11) into Equation (10) and us- ing lemma 1, we get    33243,2, 13,1,3,2, 13,1,3,2,3,1,1 0.yt ztyxzxDD DD DDDDffNNNNNNNNNNNNNNNNNN        1fNTherefore, we have shown that solves lution of Equation (2) is Equation (4) under the linear differential conditions (9), The corresponding so2,221xNNfufN  . (12) 3. Wronskian Solutions In what follows, according to [10-12], we would like to present a few special Wronskian solutions to the (3 + Open Access JAMP H. C. MA, Y. B. BAI 20 1) pinelli by-dimensional Boiti-Leon-Manna-Pemequation solving the linear conditions (9). It is well known that the correspond ing Jordan form of a real marix 1201,1m0NNJJAJ  (13) 2)  (15) where have the following two type of blocks: 1) ,iJ (14) 01ii01iiikk2220,10,,01iiiiiilliiiiiAIAJAI 0IA,ii and i have thare all real constantstype of e real eigenvalue. The first blocks i with alge- braic multiplicity and the second type of blocks have the complex eigenvalue 1,niiikkN 1iii with algebraic multiplicity 3.1. Rational Solutions Suppose A have the first type of Jordan blocks il. 11101,01NNA  (16) the eigenvalue In this case, if 10, corresponding to the following form: 0010 ,010NNA (17) from th e condition ( 9), we get  ,,,,,,0,4 ,,ixxitixxxiy ixiz,, 1.ixi  (18) where are all polynomials in 1ii,,xyz (3 + 1) and and onskian solution to the dim si (19) is called a rational Wronskian solution. From Equation (18), we get t, en-a general Wronal Boiti-Leon-Manna-Pempinelli Equation (2) 2u 112ln,,,,xkW  1,1,,1,1,1,1,0,4 ,,.xxti xxxyxzx (20) ing Maple, we get the fol- Solving Equation (20) by uslowing formulas: .Cxyz C 11 2Similarly, by solving 1,1, 1,0,4 ,i xxiti xxx 1,(21) 1,1,1,1,,,iy ixiz ixi (22) -order are ob- o be zero. 1) Zero-order: Taking corresponding Wronskian deterathen two special rational solution of lowerta l constants tined after setting some integra11Cxyz Crminant and the as2, the sociated tional Wronskian solution of zero-order read 111 2,fWCxyz C (23)  112lnxuW 122,CCxyz C   (24) where are arbitrary constants. 2) der: Taking 12,CC First-or11 2,Cxyz C we can have 232 3133 3243C222222342311 16626 2223 2223611366626.xzyxyzx tzzyzy yCxCzCCyx  Cy CzCy CCz Then, the corresponding Wronskian determinant and Wroner are Cz(25) rationalskian solution of first-ord1212221212 2,,fW P222ln ,2222,xuWCyzxyxzxyzPCC xy zCP   where 22 222223112PCxyxzxz xyzy zyxyzx33 12222 222314311422233yztCCxyyzxyxyzCxyzCCCC   Open Access JAMP H. C. MA, Y. B. BAI 21and are arbitrary real constants. Similarly, e higher order rational Wronskian so- lutions. 3.2. Solitons, Negatons and Positons 3.2.1. Solitons If 1234,,,CCCC we can obtain morA becomes to the following form 120,0NNNA (26) where the eigenvaluce 0.i nSubstituting the form of expression (26) into Equatio (9), the following system of differential equations is obtained ,4,iiiiiiiiit xxxi  (27) ,,iii iiiiyxzx  By solving system (27), we get the n-soliton solution  (28) with of Equation (2) 122ln,,,xNuW  , i being defined by 3232cosh4 ,sinh4,iiiiiiiiiixyztioddxy (29) zti even  where 120N are arbitrary constants. We and 2-soliton solutions present the 1-soliton 3211111211112lncosh4anh 4xuxyztxyzt  312t 3222ln coshxuW1111322222124 ,sinh 42xyztxyztPQ w here 3222223211111coth 4tanh 4PxyzQxyz 2ttSimilarly, we can obtain 3-soliton, 4-soliton solution and n-soliton. 3.2.2. Negatons and Positons If the eigenvalue 110, J becomes to the foll- owing form (30) We start from the eigenfuction 11111101,01kkJ 11, which is dete- rmined by 11111 11111111 1111,4,,txyxzx ,xx (31) General solution to this system in two cases of 10 are and 103211 11111cosh 4Cxyzt 323231 111341114, 0,cos 4sin 4ztxyztCxyz   2111sinhCxy1111()C211, 0,t (32) respectively, where and are arbitrary re123,,CCC 4Cal constants. When 10, wegaton solution get neand when 10, we soluTo construct Wronskian solutions corresponding to Jordan blocks of higher-order, we use the basicdeveloped for the KdV equation [10,11]. get positon tions. idea Differentiating (9) with respect to 1, we can findt the vector function tha  111111T111 1111111,,,1!1 !kk   ,(33satisfies ) 11111, 1101,01xxxkk (34) Open Access JAMP H. C. MA, Y. B. BAI Open Access JAMP 22 1, 1,1,1,4 ,txx  (35) where 11, 1,,xxxy z  denotes the tive with respect to 1deriva and k is an ar nonnegative integer. There1bitraryfore, through this set of eigenfunctions and Equation (12), a Wronskian solution of order 11k to Equation (2) is presented as:   111111112lnxW 11 111, ,,,1!1!kk  (36) orra nwhere which cesponds to the first type of Jordan blocks with onzero real eigenvalue. In what follows, several exact solutions of lower-order are presented to the (3 + 1)-dimensional Boiti-Leon- Manna-Pempinelli equation as u321111 14.xyzt, 3211 1114.xyz t   3211 14yzt 1-negaton3211111321-positon1 11132111112-negaton2lncosh2tanh4 ,2lncos42ta4,2lnxxxuxyztuxyztyz tuW       1xnx   111111321111112-positon1 1113211 11111cosh, cosh4cosh,cosh sinh122lncos, cos4cosh ,cos sin12xxyztuWxyzt   3.3. Interaction Solutions We are now presenting examples of Wronskian interac- tion solutions among different kinds of Wronskian so- lutions to the (3 + 1)-dimensional Boiti-Leon-Manna- Pempinelli equation. Let us assume that there are two sets of eigenfunctions 12 12,,,;,,kl,  (37) associated with two different eigenvalues and , ution respectively. A Wronskian sol 12 122ln, ,, ;,,xkluW2    (38) istwo solutions determined by the two sets of eigenfunc- tions in (37). In fact, we ca have more general Wron- skian interaction solutions among three or more kinds of solutions such as rational solutions, positons, solitons, negatons, breathers and complexitons. In what follows, we would like to show a few special Wronskian interaction solutions depending on rational solution, positons and solitons. Firstly, we choose three different sets of special eigenfunctions: nrational32soliton111 132positon2 222,cosh4 ,cos4 ,xyzxyztxyz twhere 10, 20 are constants. Three Wronskian interaction determinants between any two of a rational solution, a single soliton and a single p os i ton are obtained as said to be a Wronskian interaction solution between H. C. MA, Y. B. BAI 23    222soliton positon21 2121sin cos,cosh sinhcossinh ,xyzWrational soliton111rational positon,sinhcosh ,,WxyzW    where 321111 14xyzt 3221 1 114.xyz  t Further, the corresponding Wronskian interaction solutions are  rational soliton11112ln,2cosh,1oshrs xuWxyzxsinh cyz  rational positon222ln ,2cosrp xuWxyzxyz222sin cos    soliton positon211221 2122ln ,2coshcos,cosh sinhcossinhsp xuW 1   where 321111 14xyzt, 3222 2224.xyz   tThe following is one Wronskian interaction determi- nant and solution involving the three eigenfunctions. The Wronskian determinant is    rational solitonpositon,,Wxy21 1 212 21sinh cossin cosh,z  so that its corresponding Wronskian solution reads as 3rational soliton positon322ln,,,rsp xquW p      3211212 2131212111122212sinh cossin cosh,sinh sinsinh coscoshsinpxyzqxyz   2  with 321111 14xyzt, where 3222 2224.xyzt   4. Conclusion In this paper, by using the Wronskian technique, we have derived the Wronskian determinant solution for the (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation which describes the fluid propagating and can be consid-ered as a model for an incompressible fluid. Moreover, we obtained some rational solutions, soliton solutions, the resultant systems of linear partial differential equations which guarantee that the Wronskian determinant solves the equation in the bilinear form. 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