Applied Mathematics and Computation

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Alied Mathematics and Comutation 7 (0) 87 870 Contents lists available at ScienceDirect Alied Mathematics and Comutation journal homeage: wwwelseviercom/locate/amc Wronskian determinant solutions of

b Deartment of Mathematics and Statistics, University of South Florida, Tama, FL 60-5700, USA c Science Research Institute of China-North Grou Comany, Beijing 00089, PR China article info abstract

1 Alied Mathematics and Comutation 7 (0) Contents lists available at ScienceDirect Alied Mathematics and Comutation journal homeage: wwwelseviercom/locate/amc Wronskian determinant solutions of the ( + )-dimensional Jimbo Miwa equation Yaning Tang a,b,, Wen-Xiu Ma b, Wei Xu a, Liang Gao c a Deartment of Alied Mathematics, Northwestern Polytechnical University, Xi an, Shaanxi 7007, PR China b Deartment of Mathematics and Statistics, University of South Florida, Tama, FL , USA c Science Research Institute of China-North Grou Comany, Beijing 00089, PR China article info abstract Keywords: ( + )-dimensional Jimbo Miwa equation Wronskian form Rational solutions Negatons Positons A set of sufficient conditions consisting of systems of linear artial differential equations is obtained which guarantees that the Wronskian determinant solves the ( + )-dimensional Jimbo Miwa equation in the bilinear form Uon solving the linear conditions, the resulting Wronskian formulations bring solution formulas, which can yield rational solutions, solitons, negatons, ositons and interaction solutions Ó 0 Elsevier Inc All rights reserved Introduction Wronskian formulations are a common feature for soliton equations, and it is a owerful tool to construct exact solutions to the corresonding Hirota bilinear equations of the soliton equations [ 4] The resulting technique has been alied to many soliton equations such as the MKdV, NLS, derivative NLS, sine-gordon and other equations [5 0] Within Wronskian formulations, soliton solutions and rational solutions are usually exressed as some kind of logarithmic derivatives of Wronskian tye determinants [ 4] The ( + )-dimensional Jimbo Miwa equation u xxxy þ u xx u y þ u x u xy þ u yt u xz ¼ 0 ð:þ was firstly investigated by Jimbo Miwa and its soliton solutions were obtained in [5] It is the second member in the entire Kadomtsev Petviashvili hierarchy Ma [6] roosed a direct aroach to exact solutions of nonlinear artial differential equation by using rational function transformations to solve Eq () Wazwaz [7] emloyed the Hirota s bilinear method to this equation and confirmed that it is comletely integrable and it admits multile-soliton solutions of any order In [8], the traveling wave solutions of Eq () exressed by hyerbolic, trigonometric and rational functions were constructed by the G 0 /G-exansion method, where G = G(n) satisfies a second order linear ordinary differential equation A Hirota bilinear form of Eq () is D x D y þ D y D t D x D z f f ¼ f xxxy f f xxx f y f xxy f x þ f xx f xy þ f yt f f y f t f xz f þ f x f z ¼ f ðf xxxy þ f yt f xz Þþð f xxx f y f xxy f x f y f t þ f x f z Þþf xx f xy ¼ 0 ð:þ after the Cole Hof transformation u ¼ ðln f Þ x ¼ f x =f where D x, D y, D z and D t are the Hirota oerators [9] Corresonding author at: Deartment of Alied Mathematics, Northwestern Polytechnical University, Xi an, Shaanxi 7007, PR China addresses: tyaning@nwueducn (Y Tang), mawx@mathusfedu (W-X Ma) /$ - see front matter Ó 0 Elsevier Inc All rights reserved doi:006/jamc000

2 Y Tang et al / Alied Mathematics and Comutation 7 (0) In this aer, we aim to resent the Wronskian determinant solutions of the above Eq (), which will articularly lead to an aroach for constructing rational solutions and solitons to the Eq () Our results will also show the richness and diversity of solution structures of the Eq () The aer is organized as follows In Section, the Wronskian determinant solution is resented for the bilinear Eq () In Section, an aroach for constructing exact solutions including rational solutions is furnished, and many examles of solutions such as rational solutions, ositons and negatons are rovided Finally, some conclusions are given in Section 4 Wronskian formulation The Wronskian technique is introduced by Freeman and Nimmo [,0], they set / ð0þ / ðþ / ðn Þ Wð/ / / n Þ¼ðN d UÞ ¼ðN d / ð0þ / ðþ / ðn Þ Þ ¼ N P ð:þ / ð0þ N / ðþ N / ðn Þ N where U ¼ð/ / / n Þ T / ð0þ i ¼ / i / ðjþ / i j P 6 i 6 N: ð:þ j Solutions determined by u = (lnf) x with f ¼ð d N Þto the Eq () are called Wronskian solutions Proosition Assuming that / i = / i (x,y,z,t) (where t P 0, < x,y,z <, i=,,, N) has continuous derivative u to any order, and satisfies the following linear differential conditions / ixx ¼ XN j¼ k ij ðtþ/ j / iy ¼ / ix ð:4þ / iz ¼ 4/ ixxx þ / ix ð:5þ / it ¼ / ix ð:6þ then f ¼ðN d Þ defined by () solves the bilinear Eq () Before roving the above results, we state the following three known useful Lemmas ð:þ Lemma jd a bjjd c dj jjd a cjjjd b djþjd a djjd b cj ¼0 where D is N (N ) matrix, and a, b, c, d are n-dimensional column vectors ð:7þ Lemma Set a j (j =,,n) to be an n-dimensional column vector, and b j (j =,,n) to be a real constant but not to be zero Then we have X N i¼ b i ja a a N j¼ XN j¼ where ba j =(b a j,b a j,,b N a Nj ) T ja a ba j a N j ð:8þ Lemma [] Under the condition () and Lemma, the following equalities hold:! ðn d Þ XN X N k ii ðtþ k ii ðtþðn d Þ ¼ XN k ii ðtþðn d Þ! ¼½ðN d N NÞ ðn d N þ ÞŠ i¼ i¼ i¼ ¼ðN d 5 N N N NÞ ðn d 4 N N N þ Þ ðn d N N þ ÞþðN d N N þ ÞþðN d N þ Þ: ð:9þ Proof of Proosition Obviously, we always have f x ¼ðN d NÞ f xx ¼ðN d N NÞþðN d N þ Þ f xxx ¼ðN d 4 N N NÞþðN d N N þ ÞþðN d N þ Þ:

3 874 Y Tang et al / Alied Mathematics and Comutation 7 (0) Using the conditions (4) (6), we get that f y ¼ ðn d NÞ f yx ¼ ðn d N NÞþðN d N þ Þ f xxy ¼ ðn d 4 N N NÞþ6ðN d N N þ ÞþðN d N þ Þ f xxxy ¼ ðn d 5 N N N NÞþ9ðN d 4 N N N þ Þ þ 6ðN d N N þ Þþ9ðN d N N þ ÞþðN d N þ Þ f z ¼ 4ðN d 4 N N NÞ 4ðN d N N þ Þþ4ðN d N þ ÞþðN d NÞ f xz ¼ 4ðN d 5 N N N NÞ 4ðN d N N þ Þþ4ðN d N þ ÞþðN d N NÞþðN d N þ Þ f t ¼ðN d NÞ f yt ¼ ðn d N NÞþðN d N þ Þ: Hence, we have f ðf xxxy þ f yt f xz Þ¼9ðN d Þ½ ðn d 5 N N N NÞþðN d 4 N N N þ Þ þ ðn d N N þ ÞþðN d N N þ Þ ðn d N þ ÞŠ ð f xxx f y f xxy f x f y f t þ f x f z Þ¼f x ð f xxx f xxy 6f t þ f z Þ¼ 6ð d N NÞð d N N N þ Þ f xx f xy ¼ 9½ ð d N N NÞþð d N N þ Þþð d N N NÞŠ ¼ 9½ ð d N N NÞþð d N N þ ÞŠ þ 6ð d N N NÞð d N N þ Þ: Using Lemma, we obtain D x D y þ D y D t D x D z f f ¼ 9ðN d Þ½ ðn d 5 N N N NÞþðN d 4 N N N þ Þ þ ðn d N N þ ÞþðN d N N þ Þ ðn d N þ ÞŠ 6ðN d NÞðN d N N þ Þþ9½ ðn d N NÞ þðn d N þ ÞŠ þ 6ðN d N NÞðN d N þ Þ ¼6ðN d N N þ ÞðN d Þ 6ðN d NÞðN d N N þ Þþ6ðN d N NÞðN d N þ Þ ¼0: This shows that f ¼ð d N Þ solve the bilinear Eq () The corresonding solution of Eq () is u ¼ f x f ¼ ð N d NÞ ðn d Þ : Observation From the comatibility conditions / i,xxt = / i,txx (i =,,N) of the conditions () (6), we have X N j¼ k ijt ðtþ/ j ¼ 0 ði ¼ NÞ and thus it is easy to see that the Wronskian determinant W(/,/,,/ N ) becomes zero if there is at least one entry k ij satisfying k ij,t (t) 0 Observation If the coefficient matrix A =(k ij ) is similar to another matrix M under an invertible constant matrix P, ie, we have A = P MP, then e U ¼ PU solves eu xx ¼ M e U e U y ¼ e U x e U z ¼ 4 e U xxx þ e U x e U t ¼ e U x and the resulting Wronskian solutions to the Eq () are the same: uðaþ ¼@ x ln ju ð0þ U ðþ U ðn Þ j¼@ x ln jpu ð0þ PU ðþ PU ðn Þ j¼uðmþ: Based on Observation I, we only need to consider the reduced case of () (6) under da/dt = 0, ie, the following conditions: / ixx ¼ XN j¼ k ij / j / iy ¼ / jx / iz ¼ 4/ ixxx þ / ix / it ¼ / ix ð:0þ

4 Y Tang et al / Alied Mathematics and Comutation 7 (0) where A =(k ij ) is an arbitrary real constant matrix Moreover, Observation II tells us that an invertible constant linear transformation on U in the Wronskian determinant does not change the corresonding Wronskian solution, and thus, we only have to solve (0) under the Jordan form of A Wronskian solutions In rincile, we can construct general solutions of the Eq () by solving the linear conditions (0) But it is not easy In this section we will resent a few secial Wronskian solutions to the Eq () It is well known that the corresonding Jordan form of a real matrix Jðk Þ 0 Jðk Þ A ¼ Jðk m Þ nn have the following two tyes of blocks: I k i 0 k ị Jðk i Þ¼ k i k i k i II A i 0 I A i Jðk i Þ¼ A i ¼ a i b i b i a i I ¼ I A i l i l i where k i, a i and b i > 0 are all real constants The first tye of blocks have the real eigenvalue k i with algebraic multilicity k i ð P m i¼ k ffiffiffiffiffiffiffi i ¼ NÞ, and the second tye of blocks have the comlex eigenvalue k i ¼ a i b i with algebraic multilicity l i Rational solutions Suose A have the first tye of Jordan blocks Without loss of generality, let k 0 k Jðk Þ¼ : 5 0 k k k In this case, if the eigenvalue k =0,J(k ) becomes to the following form: k k from the condition (0), we get / xx ¼ 0 / iþxx ¼ / i / iy ¼ / ix / iz ¼ 4/ ixxx þ / ix / it ¼ / ix :i P : ð:þ Such functions / i (i P ) are all olynomials in x, y, z and t, and a general Wronskian solution to the ( + )-dimensional Jimbo Miwa Eq () u x ln Wð/ / / k Þ is rational and is called a rational Wronskian solution of order k

5 876 Y Tang et al / Alied Mathematics and Comutation 7 (0) From (), we solve /,xx =0,/,y =/,x, /,z =4/,xxx +/,x, /,t = /,x, and have / ¼ c ðx þ t þ y þ zþþc : Similarly, by solving / i+,xx = / i, / i+,y =/ i+, x, / i+,z =4/ i+,xxx +/ i+,x, / i+,t = / i+,x, i P, then two secial rational solutions of lower-order are obtained after setting some integral constants to be zero ) Zero-order: when c =,c =0,/ = x +y +z + t, f = W(/ )=x +y +z + t, u x ln Wð/ Þ¼ x þ y þ z þ t : ) First-order: / = x +y +z + t, ð:þ / ¼ 6 t þ t ðx þ yþþt ðx þ yþ þ 6 ðx þ yþ þ 4z þ z ðx þ y þ tþþzðx þ y þ tþ þ 4z f ¼ Wð/ / Þ¼ u x ln Wð/ / Þ¼ where ðx þ y þ z þ tþ ð:þ ¼ t þ t ðx þ yþþtðx þ yþ þ ðx þ yþ þ 8z þ 4z ðx þ y þ tþþzðx þ y þ tþ 4z: ) Second-order: / = x +y +z + t, / ¼ 6 t þ t ðx þ yþþt ðx þ yþ þ 6 ðx þ yþ þ 4z þ z ðx þ y þ tþþzðx þ y þ tþ þ 4z / ¼ t5 0 þ 5t4 t ðx þ yþþ 0 ðx þ yþ þ t ðx þ yþ þ 5t 0 ðx þ yþ4 þ ðx þ yþ5 0 þ 4z5 5 þ z4 ðx þ t þ yþþz ðx þ t þ yþ þ z ðx þ t þ yþ þ z ðx þ t þ yþ4 þ 8z þ 8z ðx þ t þ yþþzðxþt þ yþ where f ¼ Wð/ / / Þ¼ u x ln Wð/ 0 / / Þ¼ q ð:4þ ¼ 64z6 45 þ 64z5 ðx þ y þ tþþ6z4 5 ðx þ y þ tþ z4 þ z ðx þ y þ tþ 9 6z ðx þ y þ tþþ 4z ðx þ y þ tþ4 8z ðx þ y þ tþ 6z þ 4z ðx þ y þ tþ5 5 4 zðx þ y þ tþ þ t5 5 ðx þ yþþt4 ðx þ yþ þ 4t 9 ðx þ yþ þ t ðx þ yþ4 þ t 5 ðx þ yþ5 þ 45 ðx þ yþ6 x6 8y q ¼ 64z5 5 þ z4 ðx þ y þ tþþz ðx þ y þ tþ 6z 6z ðx þ y þ tþþ 4z ðx þ y þ tþ4 4zðx þ y þ tþ þ t5 5 þ t4 ðx þ yþþ4t ðx þ yþ þ 4t ðx þ yþ þ t ðx þ yþ4 þ 5 ðx þ yþ5 x5 5 : Solitons, negatons and ositons If the eigenvalue k 0, k 0 k J(k ) becomes to the following form : 0 k k k

6 Y Tang et al / Alied Mathematics and Comutation 7 (0) We start from the eigenfuction / (k ), which is determined by ð/ ðk ÞÞ xx ¼ k / ðk Þ ð/ ðk ÞÞ y ¼ ð/ ðk ÞÞ x ð/ ðk ÞÞ z ¼ 4ð/ ðk ÞÞ xxx þ ð/ ðk ÞÞ x ð/ ðk ÞÞ t ¼ð/ ðk ÞÞ x : ð:5þ General solutions to this system in two cases of k > 0 and k < 0 are ffiffiffiffiffi ffiffiffiffiffi / ðk Þ¼C sinhð k ðx þ y þ ðk þ Þz þ tþþ þ C coshð k ðx þ y þ ðk þ Þz þ tþþ k > 0 ð:6þ ffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi / ðk Þ¼C cosð k ðy þ x þ ðk þ Þz þ tþþ C 4 sinð k ðy þ x þ ðk þ Þz þ tþþ for < k < 0 ð:7þ ffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi / ðk Þ¼ C cosð k ðy þ x þ ðk þ Þz þ tþþ þ C 4 sinð k ðy þ x þ ðk þ Þz þ tþþ for k < ð:8þ resectively, where C, C, C and C 4 are arbitrary real constants But obviously the solutions (7) and (8) have the oosite sign, so we will only consider the solution (7) later By an insection, we find that / ðk Þ k 0 / ðk k / ðk Þ k / ðk Þ 6 ¼ / ðk Þ 0 k / ðk Þ and ðk k k / ðk Þ y k / ðk Þ z k / ðk Þ ¼ t xx ¼ k / ðk Þ ¼ 4 k / ðk Þ k / ðk Þ x k k þ xxx k / ðk Þ 0 6 j 6 k : x ðk k Therefore, through this set of eigenfunctions, we obtain a Wronskian solution to the Eq (): u x ln Wð/ ðk Þ k / ðk Þ ðk k / ðk ÞÞ which corresonds to the first tye of Jordan blocks with a nonzero real eigenvalue When k > 0, we get negaton solutions, and whenk < 0, we get ositon solutions If we suose A have m different nonzero real eigenvalues, in which there are l ositive real eigenvalues and m l negative real eigenvalues, then more general negaton can be obtained by combining l sets of eigenfunctions associated with different k i >0: u x ln W / ðk k / ðk Þ ðk k / ðk Þ / l ðk l k l / l ðk l Þ ðk l l k l / l ðk l Þ : Similarly, more general ositon can be obtained by combining m l sets of eigenfunctions associated with different k i <0: u x ln W / ðk k / ðk Þ ðk k / ðk Þ / m l ðk m l k m l / m l ðk m l Þ ðk m l m l k m l / m l ðk m l Þ : This solution is called an n-negaton of order (k,k,,k l ) or n-ositon of order (k,k,,k m l ) If l = n or l = 0, we simly say that it is an n-negaton of order n or an n-ositon of order n An n-soliton solution is a secial n-negaton: u x ln Wð/ / / n Þ with / i given by ffiffiffiffi / i ¼ cosh k iðx þ y þ ðki þ Þz þ tþþc i i odd ffiffiffiffi / i ¼ sinh k iðx þ y þ ðki þ Þz þ tþþc i i even where 0 < k < k < < k n and c i ( 6 i 6 n) are arbitrary real constants

7 878 Y Tang et al / Alied Mathematics and Comutation 7 (0) Two kinds of secial negatons of order k are ffiffiffi u x ln Wð/@ k k k /Þ / ¼ coshð k ðx þ y þ ðk þ Þz þ tþþcþ ffiffiffi u x ln Wð/@ k k k /Þ / ¼ sinhð k ðx þ y þ ðk þ Þz þ tþþcþ where k > 0 and c is an arbitrary constant Two kinds of secial ositons of order k are u x ln Wð/@ k k k /Þ ffiffiffiffiffiffi /ðkþ ¼cosð k ðy þ x þ ðk þ Þz þ tþþcþ k < 0 u x ln Wð/@ k k k /Þ ffiffiffiffiffiffi /ðkþ ¼sinð k ðy þ x þ ðk þ Þz þ tþþcþ k < 0: To understand the above results better, we shall give several exact solitons, ositons and negatons of lower-order as follows: () Two solitons of zero-order: ffiffiffiffiffi u x ln Wð/ Þ¼@ x lnðcoshð k ðx þ y þ ðk þ Þz þ tþþc ÞÞ ¼ ffiffiffiffiffi k tanhðh Þ ð:9þ ffiffiffiffiffi u x ln Wð/ Þ¼@ x lnðsinhð k ðx þ y þ ðk þ Þz þ tþþc ÞÞ ¼ ffiffiffiffiffi k cothðh Þ ð:0þ where h ¼ ffiffiffiffiffi k ðx þ y þ ðk þ Þz þ tþþc k > 0: One soliton of first-order: ðk k Þðsinhðh þ h Þ sinhðh h ÞÞ u x ln Wðcoshðh Þ sinhðh ÞÞ ¼ ffiffiffiffiffi ffiffiffiffiffi ffiffiffiffiffi ffiffiffiffiffi ð k k Þ coshðh þ h Þ ð k þ k Þ coshðh h Þ ð:þ where h i ¼ ffiffiffiffi k iðx þ y þ ðki þ Þz þ tþþc i, k i >0,i =, () One negaton of first-order: 4 ffiffiffiffiffi k ð þ coshðhþþ u x ln WðcoshðhÞ@ k coshðhþþ ¼ ffiffiffiffiffi k ðx þ y þ k z þ z þ tþþsinhðhþ ð:þ ffiffiffiffiffi where h ¼ð k ðx þ y þ ðk þ Þz þ tþþc Þ () Two ositon of zero-order: ffiffiffiffiffiffiffiffi u x ln Wð/ Þ¼@ x lnðcosð k ðx þ y þ ðk þ Þz þ tþþc ÞÞ ¼ ffiffiffiffiffiffiffiffi k tanðh Þ: ð:þ ffiffiffiffiffiffiffiffi u x ln Wð/ Þ¼@ x lnð sinð k ðx þ y þ ðk þ Þz þ tþþc ÞÞ ¼ ffiffiffiffiffiffiffiffi k cotðh ÞÞ: ð:4þ where h ¼ ffiffiffiffiffiffiffiffi k ðx þ y þ ðk þ Þz þ tþþc One ositon of first-order: 4 ffiffiffiffiffiffiffiffi k ð þ cosðhþþ u x ln WðcosðhÞ@ k cosðhþþ ¼ ffiffiffiffiffiffiffiffi k ðx þ y þ k z þ z þ tþþsinðhþ where h ¼ ffiffiffiffiffiffiffiffi k ðx þ y þ ðk þ Þz þ tþþc Interaction solutions ð:5þ Let us assume that there are two sets of eigenfunctions ð/ ðkþ / ðkþ / k ðkþ w ðlþ w l ðlþþ ð:6þ associated with two different eigenvalues k andl, resectively A Wronskian solution u =@ x lnw(/ (k),/ (k),,/ k (k)w (l),,w l (l)) is said to be a Wronskian interaction solution between two solutions determined by the two sets of eigenfunctions in (6) Of course, we can have more general Wronskian interaction solutions among three kinds of solutions such as rational solutions, negatons, ositons

8 Y Tang et al / Alied Mathematics and Comutation 7 (0) In what follows, we shall show a few secial Wronskian interaction solutions Let us first choose different sets of secial eigenfunctions: / rational ¼ x þ y þ z þ t ffiffiffiffiffi / soliton ¼ coshð k ðx þ y þ ðk þ Þz þ tþþ ffiffiffiffiffiffiffiffi / ositon ¼ cosð k ðx þ y þ ðk þ Þz þ tþþ where k >0,k < 0 are constants Three Wronskian interaction determinants between any two of a rational solution, a single soliton and a single ositon are Wð/ rational / soliton Þ¼ ffiffiffiffiffi k ðx þ y þ z þ tþ sinhðh Þ coshðh Þ ð:7þ Wð/ rational / ositon Þ¼ ffiffiffiffiffiffiffiffi k ðx þ y þ z þ tþ sinðh Þ cosðh Þ ð:8þ Wð/ soliton / ositon Þ¼ ffiffiffiffiffiffiffiffi k coshðh Þ sinðh Þ ffiffiffiffiffi k cosðh Þ sinhðh Þ ð:9þ where h ¼ ffiffiffiffiffi k ðx þ y þ ðk þ Þz þ tþ h ¼ ffiffiffiffiffiffiffiffi k ðx þ y þ ðk þ Þz þ tþ Further, the corresonding Wronskian interaction solutions are k ðx þ y þ z þ tþ coshðh Þ u rs x ln Wð/ rational / soliton Þ¼ffiffiffiffiffi k ðx þ y þ z þ tþ sinhðh Þ coshðh Þ ð:0þ k ðx þ y þ z þ tþ cosðh Þ u r x ln Wð/ rational / ositon Þ¼ffiffiffiffiffiffiffiffi k ðx þ y þ z þ tþ sinðh Þþcosðh Þ ð:þ ðk k Þ coshðh Þ cosðh Þ u s x ln Wð/ soliton / ositon Þ¼ffiffiffiffiffiffiffiffi k coshðh Þ sinðh Þþ ffiffiffiffiffi k sinhðh Þ cosðh Þ ð:þ where h ¼ ffiffiffiffiffi k ðx þ y þ ðk þ Þz þ tþ h ¼ ffiffiffiffiffiffiffiffi k ðx þ y þ ðk þ Þz þ tþ The following is one Wronskian interaction determinant and solution involving the three eigenfunctions ffiffiffiffiffi ffiffiffiffiffiffiffiffi Wð/ rational / soliton / ositon Þ¼ðxþyþzþtÞ k k sinhðh Þ cosðh Þþk k coshðh Þ sinðh Þ þðk k Þ coshðh Þ cosðh Þ¼ ð:þ u rs x ln Wð/ rational / soliton / ositon Þ¼ q ð:4þ where ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffi ffiffiffiffiffiffiffiffi q ¼ðxþyþzþtÞ k k ðk k Þ sinhðh Þ sinðh Þþ k k sinhðh Þ cosðh Þþk k coshðh Þ sinðh Þ h ¼ ffiffiffiffiffi k ðx þ y þ ðk þ Þz þ tþ and h ¼ ffiffiffiffiffiffiffiffi k ðx þ y þ ðk þ Þz þ tþ: When the corresonding Jordan form of a real matrix is the second tye of block, the solutions of the Eq (0) will become very comlicated, so we omit that here 4 Conclusion In sum, we gave the Wronskian determinant solutions of the ( + )-dimensional Jimbo Miwa equation through the Wronskian technique Moreover, we obtained some rational solutions, soliton solutions, ositons and negatons of this equation by solving the resultant systems of linear artial differential equations which guarantee that the Wronskian determinant solves the equation in the bilinear form All these show the richness of the solution sace of the ( + )-dimensional Jimbo Miwa equation and the resulting solutions are exected to hel understand wave dynamics in weakly nonlinear and disersive media Acknowledgments This work was suorted in art by the National Science Foundation of China under Grant No and No 000, the Established Researcher Grant and the CAS faculty develoment grant of the University of South Florida, Chunhui Plan of the Ministry of Education of China, and Wang Kuancheng Foundation One of the authors (YN Tang) would like to exress her sincere gratitude to Prof Ma for his warmest hositality and suort during her study at University of South Florida

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Research Article Two Different Classes of Wronskian Conditions to a (3 + 1)-Dimensional Generalized Shallow Water Equation

International Scholarly Research Network ISRN Mathematical Analysis Volume 2012 Article ID 384906 10 pages doi:10.5402/2012/384906 Research Article Two Different Classes of Wronskian Conditions to a 3