Applied Mathematics and Computation

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1 Applied Mathematics Computation 29 (22) Contents lists available at SciVerse ScienceDirect Applied Mathematics Computation journal homepage: wwwelseviercom/locate/amc Etended Gram-type determinant, wave rational solutions to two (3+)-dimensional nonlinear evolution equations Magdy G Asaad a,, Wen-Xiu Ma b a Department of Mathematics, Faculty of Science, Aleria University, Aleria, Egypt b Department of Mathematics Statistics, University of South Florida, Tampa, FL , USA article info abstract Keywords: Nonlinear evolution equations Jimbo Miwa equation Pfaffian Grammian determinant ilinear äcklund transformation New eact Grammian determinant solutions to two (3+)-dimensional nonlinear evolution equations are derived Etended set of sufficient conditions consisting of linear partial differential equations with variable-coefficients is presented Moreover, a systematic analysis of linear partial differential equations is used for solving the representative linear systems The bilinear äcklund transformations are also constructed for the equations Also, as an application of the bilinear äcklund transforms, a new class of wave rational solutions to the equations are eplicitly computed Ó 22 Elsevier Inc All rights reserved Introduction Many phenomena in the nonlinear sciences in the physics can be modeled by a various classes of integrable nonlinear partial differential equations Consequently, construction of wave solutions of nonlinear evolution equations plays a crucial role in the study of nonlinear sciences nonlinear phenomena To obtain the traveling wave solutions for these equations, especially for the higher-dimensional the coupled nonlinear evolution equations, can make people know the described physical phenomena esides traveling wave solutions, another set of interesting multi-eponential wave solutions [,] is a linear combination of eponential waves Nowadays, with the rapid software technology development, solving nonlinear partial differential equations via eplicit symbolic computation is taking an increasing role due to its accuracy, efficiency its restrained use To this end, in the open literature, a set of systematic methods have been developed to obtain eplicit solutions for nonlinear evolution equation, such as tanh coth function, sine cosine function, Jacobi elliptic function method, symmetry method, Weierstrass function method, the F-epansion method, Homotopy perturbation method, variational iteration method [3 33] so on However, all methods mentioned above have some restrictions in their applications On the other h, the Hirota bilinear formalism [,2] has been successfully used in the search for eact solutions of continuous discrete systems [2 22], also in the search for new integrable equations by testing for multisoliton solutions or äcklund transformations, even used in constructing N-soliton solutions for integrable couplings by perturbation [] It is now believed that most integrable systems if not all, could be transformed into bilinear forms by dependent variable transformations Therefore, one would epect to study most integrable systems within the bilinear formalism Moreover, various methods have been presented in the last four decades to construct eact solutions for many nonlinear evolution equations, such as Grammian determinant approach [,23], Wronskian determinant approach [4,5] Wronskian determinant, Grammian determinant Pfaffian solutions to the (3+)-dimensional generalized KP KP equations were constructed in [6,7] Corresponding author addresses: mgamilmailusfedu (MG Asaad), mawcasusfedu (WX Ma) 96-33/$ - see front matter Ó 22 Elsevier Inc All rights reserved

2 24 MG Asaad, WX Ma / Applied Mathematics Computation 29 (22) In this work, our aim is to investigate two well-known models that are of particular interest in science The (3+)- dimensional Jimbo Miwa equation [3]: u y þ 3u u y þ 3u u y þ 2u yt 3u z ¼ ; ðþ the (3+)-dimensional nonlinear evolution equation [6]: 3w z ð2w t þ w 2ww Þ y þ 2ðw w yþ ¼ ; ð2þ where u ¼ uð; y; z; tþ; w ¼ wð; y; z; tþ; f ¼ f Z f ¼ fd: oth of which can be written in terms of the Hirota bilinear operator Moreover, Eq () was firstly investigated by Jimbo Miwa its soliton solutions were obtained in [3] It is the second member in the entire Kadomtsev Petviashvili hierarchy Lately, Wazwaz [2,3] successfully studied one-soliton solutions to Eq () by means of the Tanh Coth method He also employed the Hirota s bilinear method to the Jimbo Miwa equation confirmed that it is completely integrable it admits multiple-soliton solutions of any order In fact Eq (2) was firstly investigated by Geng (see [6], where it was decomposed into systems of solvable ordinary differential equations with the help of the (+)-dimensional AKNS equations algebraic-geometrical solutions for it were eplicitly given in terms of the Riemann theta functions) Further, Geng Ma [4] derived an N-soliton solution of the equation its Wronskian form by using the Hirota direct method Wronskian technique Eq (2) has Grammian determinant solution in [5] Eq () has Wronskian solution in [9] In our present paper, we will show the above (3+)-dimensional nonlinear evolution equations has a class of etended Grammian determinant solutions, with all generating functions for matri entries satisfying a linear system of partial differential equations involving variable-coefficient, which guarantee that the Grammian determinant solves the equation The Jacobi identity for determinants is the key to establish the Grammian formulation [] Moreover, a systematic analysis of linear partial differential equations is used to solve the representative linear systems A theory of transformation of surfaces initiated by äcklund [24] later developed by Loewner [25] has, in recent years, proved to be of remarkable importance in the analysis of a wide range of physical phenomena, the successful applications of this transformation theory to nonlinear evolution equations have led to a rekindling of interest in this topic Perhaps the simplest eample of äcklund transformation is the Cauchy Riemann relations In particular, äcklund transformations of the Sine-Gordon equation have generated results of interest in dislocation theory [26], in the study of long Josephson junctions [27], in the investigation of propagation of long optical pulses through a resonant laser medium [2] The work by Miura [29] on the Korteweg-de Vries equation has likewise involved the use of a äcklund transformation In 95, Loewner [25] introduced an important generalization of the concept of äcklund transformation This was in connection with the reduction to canonical form of the well-known hodograph equations of gasdynamics In Section 5, we would like to present a bilinear äcklund transformation for the above (3+)-dimensional Jimbo Miwa Eq () the (3+)-dimensional nonlinear evolution Eq (2) Furthermore, we will use the transformation to generate a new class of wave rational solutions to the same equation 2 Pfaffian bilinear form 2 Pfaffian Pfaffians, which may be an unfamiliar word, are closely related to determinants They are usually defined by the property that the square of a Pfaffian is the determinant of an antisymmetric matri This feature leads often to the misundersting that a Pfaffian is a special case of a determinant In fact, it is easy to recognize that the Pfaffians are a generalization of determinants Therefore, Plücker relations Jacobi identities, which are identities for determinants, also hold for Pfaffians In this paper, we will use Pfaffian identities [3] to search for eact solutions to the nonlinear partial differential equations: () (2) Let us discuss some basics about the Pfaffian [3] Let A ¼ a be a skew-symmetric matri, in which a jk 6j;k6m j;k ¼a k;j for j; k ¼ ; 2; ; m It is known that detðaþ of odd order vanishes but detðaþ of even order m ¼ 2n is the square of a Pfaffian, that is ( ; if m is odd; detðaþ ¼ Pfða jk Þ 2 6j;k6m ; if m is even: We can denote this Pfaffian Pf ða jk Þ 6j;k62n by ð3þ Pfða jk Þ 6j;k62n ¼ð; 2; 3; ; 2nÞ: ð4þ Then, we have

3 a 2 a 3 a 4 a 2 a 23 a 24 a 3 a 23 a 34 a 4 a 24 a 34 ð; 2; 3; 4Þ 2 : Therefore, a second-order (n ¼ 2) Pfaffian is given by where MG Asaad, WX Ma / Applied Mathematics Computation 29 (22) ¼ ½a 2 a 34 a 3 a 24 þ a 4 a 23 Š 2 ð5þ ð; 2; 3; 4Þ ¼ð; 2Þð3; 4Þð; 3Þð2; 4Þþð; 4Þð2; 3Þ; ð7þ ðj; kþ ¼a j;k for j < k: ðþ Also, it should be noted that from the antisymmetric a j;k ¼a k;j, we have ð6þ ðj; kþ ¼ðk; jþ: In general, we have an epansion rule for a Pfaffian (; 2; ; 2n) of order n: ð; 2; ; 2nÞ ¼ X2n ðþ j ð; jþð2; 3; ;^j; ; 2nÞ; j¼2 ð9þ ðþ where the notation ^j means that the inde j is omitted An alternative epansion reads ð; 2; ; 2nÞ ¼ X2n ðþ j ð; 2; ;^j; ; 2n Þðj; 2nÞ: j¼ ðþ Repeating the above epansion, we arrive at the summation of products of first-order Pfaffians [3]: ð; 2; ; 2nÞ ¼ X ðþ P ði ; i 2 Þði 3 ; i 4 Þði 5 ; i 6 Þ ði 2n ; i 2n Þ; ð2þ where P means the sum over all possible combinations of pairs selected from f; 2; ; 2ng that satisfy i < i 2 ; i 3 < i 4 ; i 5 < i 6 ; i 2n < i 2n ; i < i 3 < < i 2n : These first-order Pfaffians ði; jþ are called the entries in the Pfaffian In the above equation,the factor ðþ P ¼þorif the sequence fi k g 2n k¼ is an even or odd permutation of ; 2; ; 2n Moreover, the Pfaffian ði ; i 2 ; ; i 2n Þ vanishes if i l ¼ i m for any pair of m l chosen from ; 2; ; 2n Also, the interchange of labels i l i m changes the parity of each permutation in the sum, thus, the Pfaffian has the skew-symmetric property ði ; ; i l ; ; i m ; ; i 2n Þ¼ði ; ; i m ; ; i l ; ; i 2n Þ; where 6 l < m 6 2n The Pfaffian also is denoted conventionally by Caianiello [7] a a ;2 ;3 a ;2n a 2;3 a 2;2n Pfða i;j Þ 6i;j62n ¼ ; ð4þ when n ¼, 2, the Pfaffian reads a 2n;2n Pfða i;j Þ 6i;j62 ¼ a ;2 ¼ð; 2Þ; Pfða i;j Þ 6i;j64 ¼ a ;2 a 3;4 a ;3 a 2;4 þ a ;4 a 2;3 ¼ð; 2; 3; 4Þ: Moreover, the Pfaffian obeys an epansion rule ða ; a 2 ; ; a 2N Þ ¼ X2N ða i ; a j ÞCði; jþ; 6 i 6 2N; ð5þ j¼ with the cofactor Cði; jþ being defined by Cði; jþ ¼ðÞ iþj ; i < j; a 2 ; ; ^a i ; ; ^a j ; a 2N Cðj; iþ ¼Cði; jþ; i > j; Cði; iþ ¼; where ^a k means that the label a k is omitted We have several epansion theorems on the Pfaffian elow we describe two of them, which are relevant to the present paper ð3þ

4 26 MG Asaad, WX Ma / Applied Mathematics Computation 29 (22) Proposition [] Let A be a 2n 2n skew-symmetric matri Then PfðAÞ ¼ð; 2; ; 2nÞ ¼ X sgnðrþ Yn ðrð2i Þ; rð2iþþ; r where the summation is taken over all permutations 2 2n r ¼ i i 2 i 2n with i < i 2 ; i 3 < i 4 ; ; i 2n < i 2n ; i < i 3 < < i 2n ; i¼ sgnðrþ ¼ðÞ invðrþ We have several epansion theorems on the Pfaffian elow we describe two of them, which are relevant to the present work Lemma [3] Let n be a positive integer Then ða ; a 2 ; ; 2; ; 2nÞ ¼ X2n j¼2 ðb ; b 2 ; c ; c 2 ; ; 2; ; 2nÞ ¼ X2n provided that ðb j ; c k Þ¼; for j; k ¼ ; 2: ðþ j ða ; a 2 ; ; j Þ½ð2; 3; ;^j; 2nÞþð; jþða ; a 2 ; 2; 3; ;^j; 2nÞŠ ða ; a 2 Þða ; a 2 ; ; 2; ; 2nÞ; ð7þ X 2n j¼ k¼jþ ð6þ ðþ jþk ðb ; b 2 ; j; kþðc ; c 2 ; ; 2; ;^j; ; ^k; 2nÞ; ðþ We shall use Eqs (7) () to epress the derivatives of the Pfaffain by the Pfaffians of lower order In the net lemma we describe two of the identities of Pfaffians which correspond to the Jacobi identity of determinants Lemma 2 [3] Let m n be positive integers Then ða ; a 2 ; ; a 2m ; ; 2; ; 2n Þð; 2; ; 2nÞ ¼ X2m s¼2 ðþ s ða ; a s ; ; 2; ; 2nÞða ; a 2 ; ; ^a s ; ; a 2m ; ; 2; ; 2nÞ; ð9þ ða ; a 2 ; ; a 2m 2m ; ; 2; 3; ; 2n Þð; 2; ; 2nÞ ¼ X ðþ s ða s ; ; ; 2n Þða ; a 2 ; ; ^a s ; ; a 2m ; ; ; 2nÞ: s¼ ð2þ 22 ilinear form In this subsection, we would like to transform the (3+)-dimensional nonlinear evolution Eqs () (2) into the bilinear forms by dependent variable transformations Through the dependent variable transformations: u ¼ 2ðln f Þ w ¼3ðln f Þ ; ð2þ the above (3+)-dimensional nonlinear evolution Eqs () (2) are mapped into the Hirota bilinear equations: ðd 3 D y þ 2D y D t 3D D z Þf f ¼ ; ð22þ where the bilinear differential operator D is defined by D k t Dn Dm y uðt; ; yþwðt; ; yþ ¼ ð t t Þ k ð Þ n ð y y Þ m uðt; ; yþwðt ; ; y Þj t ¼t; ¼;y ¼y: ð23þ We can rewrite Eq (22) in terms of f as follows f y þ 2f yt 3f z f f f y 3f y f þ 3f f y 2f y f t þ 3f f z ¼ : ð24þ

5 MG Asaad, WX Ma / Applied Mathematics Computation 29 (22) Sufficient conditions Grammian solutions 3 Sufficient conditions Let us introduce the following Grammian determinant: f N ¼ detða ij Þ 6i;j6N ; a ij ¼ d ij þ / i w j d; i; j ¼ ; 2; ; N; d ij ¼ ; i j ; i ¼ j ; all the elements / i ¼ / i ð; y; z; tþ w j ¼ w j ð; y; z; tþ satisfy the linear differential equations: / i;y ¼ 2nðtÞ/ i; ; / i;z ¼ 2nðtÞ/ i; þ XN k ik ðtþ/ k ; k¼ / i;t ¼ / i; þ XN ðtþ/ k ; >< k¼ w j;y ¼2nðtÞw j; ; w j;z ¼2nðtÞw j; þ XN ðtþw l ; l¼ w j;t ¼ w j; þ XN ðtþw l ; l¼ where nðtþ, k ik ðtþ, ðtþ, ðtþ ðtþ are arbitrary differentiable functions in t, k ij ðtþþl ji ðtþ ¼, g ij ðtþþq ji ðtþ ¼ for i; j ¼ ; 2; ; N In what follows, as an application of the Pfaffian technique, we shall construct new Grammian solutions to the (3+)-dimensional Jimbo Miwa Eq () the (3+)-dimensional nonlinear evolution Eq (2) Theorem 3 (Sufficient condition) Let / i ð; y; z; tþ w j ð; y; z; tþ, i; j ¼ ; 2; ; N, satisfy (27), then the Grammian determinant f N defined by (25) solves the Hirota bilinear Eq (22) the functions u ¼ 2ðln f N Þ w ¼3ðln f N Þ solves the (3+)- dimensional Jimbo Miwa Eq () the (3+)-dimensional nonlinear evolution Eq (2) respectively ð25þ ð26þ ð27þ Proof Let us epress the Grammian determinant f N by means of a Pfaffian as f N ¼ð; 2; ; N; N ; ; 2 ; Þ ¼ ðþ; where ði; j Þ¼a ij ði; jþ ¼ði ; j Þ¼ To compute the derivatives of the entries ði; j Þ the Grammian f N, we introduce new Pfaffian entries ð2þ ðd n ; j Þ¼ n w j; ðd n n n ; iþ ¼ / i; n ðd m ; d n Þ¼ðd n; iþ ¼ðd m ; j Þ¼; m; n P ; ð29þ by using Eqs (26) (27), we can get ði; j Þ¼/ i w j ¼ðd ; d ; i; j Þ; ð3þ Z y ði;j Þ¼ ð/ i;yw j þ/ i w j;yþd ¼ 2nðtÞ ð/ i;w j / i w j;þd ¼ 2nðtÞð/ i;w j / i w j;þ¼2nðtþ½ðd ;d ;i;j Þþðd ;d ;i;j ÞŠ; ð3þ Z t ði; j Þ¼ ð/ i;tw j þ / i w j;tþd ¼ ¼ ½/ i;w j þ / i w j;šd þ XN "!!# / i; þ XN / k w j þ / i w j; þ XN w l d k¼ ¼ð/ i; w j / i; w j; þ / i w j; Þþ XN a kj þ XN a il g ij q ji k¼ Z d kj þ / k w j d þ XN d il þ / i w l d XN d kj XN d il k¼ ¼½ðd 2 ; d ; i; j Þðd ; d ; i; j Þþðd ; d 2 ; i; j ÞŠ þ XN a kj þ XN a il ; l¼ k¼ l¼ l¼ l¼ k¼ l¼ ð32þ

6 2 MG Asaad, WX Ma / Applied Mathematics Computation 29 (22) z ði; j Þ¼ þ XN "!# "!# 2n/ i; þ XN k ik / k w j þ / i 2nw j; þ XN w l d ¼ 2nðtÞZ ð/ i; w j / i w j; Þd k¼ k¼ k ik d kj þ / k w j d þ XN l¼ l¼ d il þ / i w l d XN ¼ 2nðtÞð/ i; w j / i; w j; þ / i; w j; / i w j; Þþ XN k ik a kj þ XN k¼ k¼ k ik d kj XN d il l¼ l¼ a il k ij l ji ¼ 2nðtÞ½ðd ; d 3 ; i; j Þðd ; d 2 ; i; j Þþðd 2 ; d ; i; j Þðd 3 ; d ; i; j ÞŠ þ XN k ik a kj þ XN a il : Therefore, from the above results (3) (33), we have the following differential formulae for f N : f N; ¼ d ; d ; ; ð34þ f N;y ¼ 2n½ðd ; d ; Þ þ ðd ; d ; ÞŠ; ð35þ f N;t ¼ðd 2 ; d ; Þ ðd ; d ; Þ þ ðd ; d XN 2 ; Þ þ ðg ll þ q ll ÞðÞ; ð36þ l¼ f N;z ¼ 2n½ðd ; d 3 ; Þ ðd ; d 2 ; Þ þ ðd 2; d ; Þ ðd 3; d XN ; ÞŠ þ ðk ll þ l ll ÞðÞ; ð37þ l¼ f N; ¼ðd ; d ; Þ þ ðd ; d ; Þ; ð3þ f N;y ¼ 2n½ðd 2 ; d ; Þ þ ðd ; d 2 ; ÞŠ; ð39þ f N; ¼ðd 2 ; d ; Þ þ 2ðd ; d ; Þ þ ðd ; d 2 ; Þ; ð4þ f N;y ¼ 2n½ðd ; d 3 ; Þ þ ðd ; d 2 ; Þ ðd 2; d ; Þ ðd 3; d ; ÞŠ; ð4þ f N;z ¼ 2n½ðd ; d 4 ; Þ ðd 4; d ; Þ þ ðd 2; d ; d ; d ; Þ ðd ; d 2 ; d ; d XN ; ÞŠ þ ðk ll þ l ll Þ d ; d ; l¼ ¼ 2n½ðd ; d 4 ; Þ ðd 4; d ; Þ þ ðd 2; d ; d ; d ; Þ ðd ; d 2 ; d ; d ; ÞŠ; ð42þ f N;yt ¼ 2n½ðd ; d 4 ; Þ ðd 4; d ; Þ þ ðd 3; d ; Þ ðd ; d 3 ; Þ þ ðd 2; d ; d ; d ; Þ ðd ; d 2 ; d ; d ; ÞŠ k¼ l¼ ð33þ þ 2n XN ðg ll þ q ll Þðd ; d ; Þ 2nXN ðg ll þ q ll Þðd ; d ; Þ l¼ l¼ ¼ 2n½ðd ; d 4 ; Þ ðd 4; d ; Þ þ ðd 3; d ; Þ ðd ; d 3 ; Þ þ ðd 2; d ; d ; d ; Þ ðd ; d 2 ; d ; d ; ÞŠ; f N;y ¼ 2n½ðd ; d 4 ; Þ ðd 4; d ; Þ 2ðd 3; d ; Þ þ 2ðd ; d 3 ; Þ þ ðd 2; d ; d ; d ; Þ ðd ; d 2 ; d ; d ; ÞŠ; ð43þ ð44þ where we have used the abbreviated notation ¼; 2; ; N; N ; ; 2 ; We can now compute that f N;y þ 2f N;yt 3f N;z f ¼ 2n 2 ðd 2; d ; d ; d ; Þ 2 ðd ; d 2 ; d ; d ; Þ 2 ðd 2; d ; d ; d ; Þ þ 2 ðd ; d 2 ; d ; d ; Þ ðþ ¼ 2n½ðd 2 ; d ; d ; d ; Þ ðd ; d 2 ; d ; d ; ÞŠðÞ; ðf N; f y 3f N;y f 2f N;y f N;t þ 3f N; f N;z Þ ¼ 2n 2 ðd 2; d ; Þðd ; d ; Þ 2 ðd 2; d ; Þðd ; d ; Þ þ 2 ðd ; d 2 ; Þðd ; d ; Þ 2 ðd ; d 2 ; Þðd ; d ; Þ þ d ; d ; ðd2 ; d ; Þ d ; d ; ðd ; d 2 ; Þ ; ð3f N; f N;y Þ¼2n 2 ðd 2; d ; Þðd ; d ; Þ þ 2 ðd ; d 2 ; Þðd ; d ; Þ 2 ðd 2; d ; Þðd ; d ; Þ þ 2 ðd ; d 2 ; Þðd ; d ; Þ : Substituting the above derivatives of f N into the LHS of Eq (24), we arrive at f N;y þ 2f N;yt 3f N;z f fn; f y 3f N;y f þ 3f N; f N;y 2f N;y f N;t þ 3f N; f N;z ¼ 2n½ðd ; d ; d 2; d ; ÞðÞ d ; d ; ðd2 ; d ; Þ ðd ; d 2; Þðd ; d ; ÞŠ 2n½ðd ; d ; d ; d 2 ; ÞðÞ d ; d ; ðd ; d 2 ; Þ ðd ; d ; Þðd ; d 2 ; ÞŠ ¼ ; ð45þ where we have made use of the known Jacobi identities for determinants This shows that the Grammian determinant f N ¼ ðþ, with the conditions (27) solves the Hirota bilinear Eq (22) the functions u ¼ 2ðln f N Þ w ¼3ðln f N Þ solves the (3+)- dimensional Jimbo Miwa Eq () the (3+)-dimensional nonlinear evolution Eq (2) respectively This ends the proof h

7 MG Asaad, WX Ma / Applied Mathematics Computation 29 (22) From the above theorem, we can see that if a set of functions / i ð; y; z; tþ w j ð; y; z; tþ,6 i; j 6 N, satisfy the conditions (27), then the determinant f N ¼ detða ij Þ is an eact solution to the Hirota bilinear Eq (22), where the entry a ij is defined in (26) The conditions (27) consist of three linear systems of second-order, third-order fourth-order partial differential equations It is rather difficult to solve those linear systems eplicitly, but based on the idea used in [9], we can find some special solutions to the present linear systems efore we proceed to solve (27), let us observe the Grammian determinants solutions more carefully Remark 4 From the compatibility conditions / i;yt ¼ / i;ty,6 i 6 N, we have the equality 2n t ðtþ/ i; ¼ ; 6 i 6 N: ð46þ Therefore, in the following discussion we assume that nðtþ ¼n Remark 5 From the compatibility conditions / i;zt ¼ / i;tz,6 i 6 N, we have the equality X N k¼ k ik;t / k ¼ ; 6 i 6 N: ð47þ Therefore, we note that if there is one entry k ij satisfying k ij;t, then / j ¼ So in the following discussion, we assume that the coefficient matri K ¼ðk ij Þ is a real constant matri Remark 6 If the coefficient matri K ¼ðk ij Þ is similar to another matri H ¼ðh ij Þ under an invertible constant matri P, that is P KP ¼ H, if we take the notation U ¼ð/ ; / 2 ; ; / N Þ T, then U ¼ P U solves U y ¼ 2n U ; U z 2n U ¼ HU; U t ¼ U : ð4þ The coefficient matri H ¼ðh ij Þ NN is equal to K T If we take the notation W ¼ðw ; w 2 ; ; w N Þ T, then W ¼ P T W solves W y ¼ 2n W ; W z 2n W ¼H T W; W t ¼ W : ð49þ We can rewrite the Grammian matri (26) to I þ A, where I is the identity matri, the element of matri A is Aði; jþ ¼ R / i w j d Noting that U ¼ P U W T ¼ W T P, we have the following result: detðd ij þ / i w j dþ NN ¼ detði þ AÞ ¼detðP ði þ AÞPÞ ¼detðI þ / i w j dþ: It follows that the resultant Grammian determinant solutions to the (3+)-dimensional Jimbo Miwa Eq () the (3+)- dimensional nonlinear evolution Eq (2) keep the same under the similar transformation Therefore, based on the above Remarks, in order to construct Grammian determinant solutions to the (3+)-dimensional nonlinear partial differential Eq (), we only need to consider the reduced case of (27) under n ¼ =2, g ij ¼ q ij ¼ dk=dt ¼, ie, the following conditions: / i;y ¼ / i; ; / i;z ¼ / i; þ XN k ik / k ; / i;t ¼ / i; ; >< k¼ w j;y ¼w j; ; w j;z ¼w j; þ XN ð5þ w l ; w j;t ¼ w j; ; l¼ k ij þ l ji ¼ ; 6 i; j 6 N; where k ij are arbitrary real constants On the other h, the Jordan form of a real matri has the following types of the blocks: k i k i C A k i N i I 2 N i C A k i k i ; ð5þ ; N i ¼ a i b i b i a i ; I 2 ¼ ; ð52þ I 2 N i l i l i where k i, a i b i > are all real constants The first type of blocks has p the ffiffiffiffiffiffiffi real eigenvalue k i with algebraic multiplicity k i, the second type of blocks has the comple eigenvalues k i ¼ a i b i with algebraic multiplicity li We will construct

8 22 MG Asaad, WX Ma / Applied Mathematics Computation 29 (22) solutions for the system of differential equations defined by (5), according to the situations of eigenvalues of the coefficient matri ased on Remark 454, all we need to do is to solve a group of subsystems in the sufficient condition on the Grammian determinant solutions, whose coefficient matries are of the forms (5) (52) 4 Some solutions for the representative systems In this section, we would like to construct some solutions to the associated system of differential equations defined by (5) For a nonzero real eigenvalue k i, we start from the eigenfunction determined by >< ðþ y ¼ðÞ ; ðþ z ðþ ¼ k i ; ð53þ ðþ t ¼ðÞ : Two special solutions to this system in two cases of k i > k i < are ¼e 4 d2 i ð2yþðþd2 i ÞzÞ ðc i sin # þ C 2i cos #Þ; p d i ¼ ffiffiffiffi k i; ¼e 4 d2 i ð2yðþd2 i ÞzÞ ðc i sinh # þ C 2i cosh #Þ; p d i ¼ ffiffiffiffiffiffiffi k i; where # ¼ p ffiffiffi d i þ p 2 2 ffiffiffi d 3 i 2 t; C i C 2i are arbitrary real constants y an inspection, we find that k i z 4! k i k i! k i ¼ C A C ; A C A k i ðk i Þ! k i k i j! j k i ¼ y j! j k i ¼ t j! j k i j! j k i ; 6 j 6 k i ; ; 6 j 6 k i ; k i k i ðk i Þ! k i k i where ki denotes the derivatives with respect to k i k i is an arbitrary non-negative integer Noting that the coefficient matri ðl ij Þ of w i satisfies l ij þ k ij ¼, we take >< ðw i ðk i ÞÞ y ¼ðw i ðk i ÞÞ ; ðw i ðk i ÞÞ z þðw i ðk i ÞÞ ¼k i w i ðk i Þ; ð54þ ðw i ðk i ÞÞ t ¼ðw i ðk i ÞÞ : Two special solutions to this system in two cases of k i > k i < are p w i ðk i Þ¼e 4 d2 i ð2yðþd2 i ÞzÞ ðd i sin # þ D 2i cos #Þ; d i ¼ ffiffiffiffi k i; p w i ðk i Þ¼e 4 d2 i ð2yþðþd2 i ÞzÞ ðd i sinh # þ D 2i cosh #Þ; d i ¼ ffiffiffiffiffiffiffi k i; where # ¼ p ffiffiffi d i þ p 2 2 ffiffiffi d 3 i 2 t; D i D 2i are arbitrary real constants Then we can find that ðk i Þ! k i k i w i ðk i Þ k i ð z þ 4 Þ k i ¼ C! k i w i ðk i Þ A C A w i ðk i Þ k i k i k i ðk i Þ! k i k i! k i w i ðk i Þ w i ðk i Þ w i ðk i Þ C A

9 MG Asaad, WX Ma / Applied Mathematics Computation 29 (22) Therefore, through these two sets of eigenfunctions, we can construct Grammian determinant entry then obtain solutions to the bilinear Eq (22) For the second type of Jordan blocks of the coefficient matri, we set a pair of eigenfunctions: U i ða i ; b i Þ¼ð/ i ða i ; b i Þ; / i2 ða i ; b i ÞÞ T ; is determined by U iz U i ¼ NU i ; U i ¼ / iða i ; b i Þ / i2 ða i ; b i Þ / ij ða i ; b i Þ ¼ / y ijða i ; b i Þ ; ; N i ¼ a i b i b i a i / ijða i ; b i Þ ¼ / t ijða i ; b i Þ Again by inspection, one can see that U i N i z 4! a i U i I 2 N i ¼ C A C A U i I 2 N i Similarly, if ðl i Þ! k i a i W i ða i ; b i Þ¼ðw i ða i ; b i Þ; w i2 ða i ; b i ÞÞ T ; is determined by W iz þ W i ¼NW i ; W i ¼ w iða i ; b i Þ w i2 ða i ; b i Þ w ij ða i ; b i Þ ¼ w y ijða i ; b i Þ ; l i l i ; N i ¼ a i b i b i a i w ijða i ; b i Þ ¼ w t ijða i ; b i Þ Again by inspection, one can see that W i N i I 2 z þ 4! a i W i N i ¼ C A C I 2 A ðl i Þ! k i a i W i N i ; ð55þ ; j ¼ ; 2: U i! a i U i ðl i Þ! k i a i U i l i l i : C A ; ð56þ ; j ¼ ; 2: W i! a i W i ðl i Þ! k i a i W i : C A One special solution to Eqs (55) (56) are given as follows / i ¼ epð þ y þ t þðþa i ÞzÞ cosðb i zþ; >< / i2 ¼ epð þ y þ t þðþa i ÞzÞ sinðb i zþ; w i ¼ epð þ y þ t ðþa i ÞzÞ cosðb i zþ; w i2 ¼epð þ y þ t ðþa i ÞzÞ sinðb i zþ; where the parameters a i b i are arbitrary constants ð57þ 5 ilinear äcklund transformation 5 äcklund transformation The äcklund transformations are essentially defined as a pair of partial differential relations involving two independent variables their derivatives which together imply that each one of the dependent variables satisfies separately a partial differential equation Thus, for eample, the transformation L ðu; u t ; u ; u y ; u ; Þ L 2 ðv; v t ;v ; v y ;v ; Þ; would imply that u v satisfy partial differential equations of the operational form, PðuÞ ¼ QðvÞ ¼: In this paper we would like to present a bilinear äcklund transformation for the above (3+)-dimensional nonlinear evolution equations () (2) Let us suppose that we have another solution f to the generalized bilinear Eq (22):

10 222 MG Asaad, WX Ma / Applied Mathematics Computation 29 (22) ðd 3 D y þ 2D y D t 3D D z Þf f ¼ ; ð5þ then we introduce the key function P ¼½ðD 3 D y þ 2D y D t 3D D z Þf f Šf 2 ½ðD 3 D y þ 2D y D t 3D D z Þf f Šf 2 : ð59þ If P ¼ then f solves the bilinear Eq (22) if only if f solves the bilinear Eq (22), that is f is a solution of ð22þ () f is a solution of ð22þ: Therefore, if we can obtain, from P ¼ by interchanging the dependent variables f f, a system of bilinear equations that guarantees P ¼ : F ðd t ; D ; D y ; D z Þf f ¼ ; F 2 ðd t ; D ; D y ; D z Þf f ¼ ; F M ðd t ; D ; D y ; D z Þf f ¼ ; where the F is are polynomials in the indicated variables M is natural number depending on the compleity of the equation It is known that the identities of Hirota s bilinear operators are necessarily to split P into a system of polynomials F i s Let us now introduce the following useful identities for Hirota s bilinear operators: D g ðd f a bþba ¼ D f ðd g a bþba; D g ab cd ¼ðD g a dþcb adðd g c bþ; b 2 ðd 2 f a aþðd2 f b bþa2 ¼ 2D f ðd f a bþba; b 2 ðd g D f a aþðd g D f b bþa 2 ¼ 2D f ðd g a bþba; ð6þ ð6þ ð62þ ð63þ b 2 ð2d 3 g D fa aþa 2 ð2d 3 g D fb bþ ¼D g ð3d 2 g D fa bþba þ D g ð3d 2 g a bþðd fb aþþd g ð6d g D f a bþðd g b aþ þ D f ðd 3 g a bþba þ D fð3d 2 g a bþðd gb aþ: ð64þ Noting that, the above identities (6) (63) can be found in [], the identity (64) can be obtained by making the independent variable transformation D g! D g þ D f in the bilinear identity [] b 2 ðd 4 g a aþa2 ðd 4 g b bþ ¼2D g½ðd 3 g a bþba þ 3ðD2 g a bþðd gb aþš; comparing the coefficient of In the above identities a, b are arbitrary continuous functions of the independent variables g, f For more identities general echange formulas you may see [] Applying the above identities on Eq (59) we can obtain f 2 ð4d y D t f f Þf 2 ð4d y D t f f Þ¼D y ðd t f f Þf f ; f 2 ð6d D z f f Þþf 2 ð6d D z f f Þ¼2D ðd z f f Þf f ; ð65þ ð66þ f 2 ð2d 3 D yf f Þf 2 ð2d 3 D yf f Þ¼D ð3d 2 D yf f Þf f þ D ð3d 2 f f ÞðD y f f ÞþD ð6d D y f f ÞðD f f Þ þ D y ðd 3 f f Þf f þ D y ð3d 2 f f ÞðD f f Þ: ð67þ Substituting the above results into the right-h side of Eq (59) we may obtain 2P ¼ D y ðd t f f Þf f þ D y ðd 3 f f Þf f 2D ðd z f f Þf f þ D ð3d 2 D yf f Þf f þ D ð3d 2 f f ÞðD y f f Þ þ D ð6d D y f f ÞðD f f ÞþD y ð3d 3 f f ÞðD f f Þ: ð6þ Lemma 7 Let f f be arbitrary continuous functions of independent variables ; y; z; t Then D ðd z f f ÞðD f f Þ¼ D z ðd 2 f f Þff D ðd D z f f Þff Let us now introduce new arbitrary parameters k, l, n, # e i,(i¼; 2; 3), into Eq (6) to obtain 2P ¼ D y ½ðD t þ D 3 ld2 kd e Þf f Šf f þ D ½ð3D 2 D y 2D z ld D y kd y e 2 Þf f Šf f þ D ½ð3D 2 #D y e 3 Þf f ŠðD y f f ÞþD ½ð6D D y ld y Þf f ŠðD f f ÞþD y ½ð3D 2 nd e 3 Þf f ŠðD f f Þ: ð69þ This is possible because the coefficients of k, l, n, # e i,(i ¼ ; 2; 3),

11 MG Asaad, WX Ma / Applied Mathematics Computation 29 (22) k : D y ½D f f Šf f D ½D y f f Šf f ; l : D ðd z f f ÞðD f f ÞD z ðd 2 f f Þff D ðd D z f f Þff n : D y ½D f f ŠðD f f Þ; # : D ½D y f f ŠðD y f f Þ; e : D y ½ff Šf f ; e 2 : D ½ff Šf f ; e 3 : D ½ff ŠðD y f f ÞD y ½ff ðd f f Þ; are all equal to zero because of Lemma 5, the properties D f f f ¼ ; D f f g ¼D f g f ; D g ðd f f gþgf ¼ D f ðd g f gþgf : ð7þ ð7þ ð72þ Then P ¼ iff i f f ¼ ; 6 i 6 5, where F is can be found from Eq (69) as follows F f f ðd t þ D 3 ld2 kd e Þf f ¼ ; >< F 2 f f ð3d 2 D y 2D z ld D y kd y e 2 Þf f ¼ ; F 3 f f ð3d 2 #D y e 3 Þf f ¼ ; F 4 f f ð6d D y ld y Þf f ¼ ; F 5 f f ð3d 2 nd e 3 Þf f ¼ : Since the coefficients of k, l, n, #, e, e 2 e 3 are zero because of Eqs (7) (72) Lemma 5, this shows that Eqs (73) presents a äcklund transformation for the (3+)-dimensional nonlinear evolution Eqs () (2) 52 Traveling wave solutions In what follows, as an application of the bilinear äcklund transformation (73), we shall construct new solutions to the (3+)-dimensional soliton Eqs (2) () For this purpose, we start with f ¼, which is the trivial solution of Eq (22) obviously Noting that D n f / ¼ n f n /; n P ; ð74þ then, the bilinear äcklund transformation (73) associated with f ¼ becomes a system of linear partial differential equations f t þ 3 f 3 l 2 f 2 k f e f ¼ ; 3 3 f f 2 l 2 f k f e 2 y z y y 2f ¼ ; >< 3 2 f # f e 2 y 3f ¼ ; 6 2 f l f ¼ ; y y 3 2 f n f e 2 3f ¼ : Let us consider a class of eponential wave solutions of the form f ¼ þ ee kþlyþmztþf ; f ¼ const; ð76þ where e, k, l, m are constants to be determined Upon selecting e ¼ ; e 2 ¼ ; e 3 ¼ : ð77þ After tedious but straightforward calculations we get ( m ¼ 2 lð3k2 kþ; ¼ kð7k2 kþ; l ¼6k; n ¼3k; # ¼3 k2 l : Therefore, we obtain a class of eponential wave solutions to the (3+)-dimensional bilinear Eq (22): ð73þ ð75þ ð7þ f ¼ þ e ep k þ ly lð3k2 kþ 2 z kð7k2 kþ t þ f!; ð79þ

12 224 MG Asaad, WX Ma / Applied Mathematics Computation 29 (22) where e, k, l, k f are arbitrary constants; u ¼ 2ðln f Þ ; solves the (3+)-dimensional Jimbo Miwa Eq (), w ¼3ðln f Þ ; ðþ ðþ solves the (3+)-dimensional nonlinear evolution Eq (2) Let us second consider a class of first-order polynomial solutions f ¼ k þ ly þ mz t; where k, l, m are constants to be determined Similarly Upon selecting e ¼ ; e 2 ¼ ; e 3 ¼ ; ð3þ a direct computation shows that the system (75) becomes >< kk ¼ ; 2m kl ¼ ; #l ¼ ; ll ¼ ; nk ¼ : Again after straightforward calculations we get m ¼ 2 kl; ¼ kk: ð5þ 2 Therefore, we obtain a class of solutions to the (3+)-dimensional bilinear Eq (22): f ¼ k þ ly kl 2 z 2 kkt þ f ; ð6þ where k, l, k f are arbitrary constants; u ¼ 2ðln f Þ ¼ 2k k þ ly þ kl z ; ð7þ kkt þ f 2 2 produces a class of rational solutions to the (3+)-dimensional Jimbo Miwa Eq (), w ¼3ðln f Þ ¼ 3k 2 k þ ly kl z 2 ; ðþ kkt þ f 2 2 produces a class of rational solutions to the (3+)-dimensional nonlinear evolution Eq (2) 6 Conclusions remarks We have built an etended Grammian formulation for the (3+)-dimensional nonlinear evolution equations: u y þ 3u u y þ 3u u y þ 2u yt 3u z ¼ ; 3w z ð2w t þ w 2ww Þ y þ 2ðw w yþ ¼ ; The facts used in our construction are the Jacobi identity for determinants Theorem 3 presents the main results on Grammian solutions, which say that u ¼ 2 ðln f NÞ; f N ¼ detða ij Þ 6i;j62N ; ð2þ ð4þ w ¼3 2 2 ðln f NÞ; f N ¼ detða ij Þ 6i;j62N ; where the elements of f N are defined by a ij ¼ d ij þ R / i w j d, i; j ¼ ; 2; ; 2N, with / i w j satisfying / i;y ¼ 2n/ ð2þ i ; / i;z ¼ 2n/ ð4þ i þ X2N k¼ k ik / k ; / i;t ¼ / ð3þ i þ X2N / k ; k¼ w j;y ¼2nw ð2þ j ; w j;z ¼2nw ð4þ j þ X2N l¼ w l ; w j;t ¼ w ð3þ j þ X2N w l ; l¼

13 MG Asaad, WX Ma / Applied Mathematics Computation 29 (22) where n, k, l, g q are arbitrary continuous function in t, solves the above (3+)-dimensional nonlinear evolution equations In Theorem 3, we only considered specific sufficient conditions: (27), though, there is a free set of continuous functions in the conditions In Section 4, we constructed some solutions for the representative systems of sufficient conditions (27), The conditions (27) are a generalization to one given in [5], which deal with Eq (2) only, but in this work, we deal with both Eqs () (2) Actually if nðtþ ¼=2 the coefficient matri ¼ðk ij Þ the coefficient matri ¼ðg ij Þ are zero, then the result of the above theorem boils down to result in [5] on the study of Eq (2) The bilinear äcklund transformations were furnished for the (3+)-dimensional nonlinear evolution Eqs () (2), based on the eistence of echange identities for Hirota bilinear operators In Section 52, We constructed a new class of eact wave solutions a new class of rational solutions to the above (3+)-dimensional nonlinear evolution Eqs () (2) of the forms respectively References u ¼ 2 ðln f 2 Þ w ¼3 ðln f 2 Þ; 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