Bilinear Bäcklund transformations and explicit solutions of a (3 + 1)-dimensional nonlinear equation

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1 Hu and Tao Advances in ifference Equations :1 OI /s R E S E A R C H Open Access Bilinear Bäcklund transformations and explicit solutions of a + 1-dimensional nonlinear equation Zhigang Hu * and Xiuli Tao * Correspondence: xzhzgya@16.com College of Science, China University of Mining and Technology, Xuzhou, 1116, P.R. China Abstract Binary Bell polynomials are applied to construct two kinds of bilinear derivative equations of a + 1-dimensional nonlinear equation. Based on one of the bilinear forms, we derive a Bäcklund transformation, the corresponding Lax pair, infinite conservation laws, and explicit solutions with an arbitrary function in y.inthe meantime, from the other bilinear form, we get another bilinear Bäcklund transformation and exact solutions by utilizing the exchange formulas for Hirota s bilinear operators. PACS Codes: Yv; 0.0.Jr Keywords: binary Bell polynomial; solitary wave solution; Bäcklund transformation 1 Introduction Nonlinear evolution equations NLEEs have attracted intensive attention in the past few decades, since they can model many important phenomena and dynamic processes in physics, mechanics, chemistry, and biology [1, ]. The study ofsolutions fornlees is very important in nonlinear physical phenomena and many effective methods have been used. Numerical methods include the decomposition method [], the iteration method [4], and the spectral method [5, 6] are developed in recent years. Various effective analytic methods have been used to explore different kinds of solutions of NLEEs, such as the inverse scattering method [7], the arboux transformation [8, 9], the Bäcklund transformation [10 1], the Hirota method [1, 14], the algebra-geometric method [15 17], the homotopy analysis method [18], and so on. Among the above methods, the Hirota method is a powerful and direct approach to construct exact solutions of NLEEs. Once a nonlinear equation is written in bilinear form, its multi-soliton solutions and rational solutions are usually obtained in a systematic way. Unfortunately, this method relies on particular skills, complex calculation, and suitable variable transformation. In recent years, Lambert, Gilson et al. proposed an effective method based on the use of the Bell polynomials to obtain bilinear Bäcklund transformation and Lax pair for soliton equations in a direct way [19 1]. Fan developed this method to find bilinear Bäcklund transformations, Lax pairs, infinite conservation laws of nonisospectral and variable-coefficient KdV, KP equations [, ]. Wang applied the binary Bell polynomials to construct bilinear forms, bilinear Hu and Tao 016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Hu and Tao Advances in ifference Equations :1 Page of 1 Bäcklund transformations, Lax pairs of a generalized + 1-dimensional Korteweg-de Vries equation, and the modified generalized Vakhnenko equation [4, 5]. The Bäcklund transformation has been proven to be another powerful approach to obtain explicit solutions of NLEEs. It is well known that the Bäcklund transformation is a transformation that is related to a pair of solutions of a nonlinear equation. The exchange property of Hirota s bilinear operators is found to be an effective tool in deriving a Bäcklund transformation for a given equation [14]. Wu recently constructed a bilinear Bäcklund transformation for a + 1-dimensional soliton equation by using the Hirota bilinear method and the exchange property of the -operators [6]. Ma extended this method to a + 1-dimensional generalized KP equation, and obtained its bilinear Bäcklund transformation and explicit solutions [7]. In this paper, we would like to study the following + 1-dimensional nonlinear equation: u xxxy u x u y x u yt +u yz = If we take z = t, equation1.1 is reduced to the Boiti-Leon-Manna-Pempinelli equation [8 ] u xxxy + u yt u xx u y u x u xy =0. 1. So, equation 1.1 is a generalization of the BLMP equation. The main goal of this paper is twofold. First of all, we will apply the binary Bell polynomials to construct two bilinear forms of equation 1.1. Further, from one of the bilinear forms, we will derive a Bäcklund transformation, the corresponding Lax pair, infinite conservation laws, and explicit solutions of equation 1.1. Second, based on the other bilinear form, we would like to construct another bilinear Bäcklund transformation and exact solutions of equation 1.1 by utilizing the exchange formulas for Hirota s bilinear operators. This paper is organized as follows. In Section, we give a brief introduction of the binary Bell polynomials. In Section, we construct two bilinear forms of equation 1.1. Further, based on one of the bilinear forms, exact solutions are obtained by using the Hirota method. In Section 4, the bilinear Bäcklund transformation, the corresponding Lax pair, and infinite conservation laws are obtained by using the binary Bell polynomials. In Section 5, another bilinear Bäcklund transformation and traveling wave solutions of equation 1.1 are derived by using the exchange formulas of Hirota s bilinear operators. Finally, someconclusionsaregiveninthelastsection. Binary Bell polynomials To begin with, we will briefly introduce some basic concepts and notations of the Bell polynomials. For details, refer to [19 1]. Let f = f x 1, x,...,x n beac function with n independent variables, the multidimensional Bell polynomials Y -polynomials are defined as follows: Y n1 x 1,...,n l f =Y n1,...,n l f r1 x 1,...,r l = exp f n 1 x 1 n l expf,.1

3 Hu and Tao Advances in ifference Equations :1 Page of 1 where f r1 x 1,...,r l = r 1 x 1 r l f, r 1 =0,...,n 1, r l =0,...,n l. Based on the multi-dimensional Bell polynomials, the multi-dimensional binary Bell polynomials Y-polynomials can be defined as follows: Y n1 x 1,...,n l v, w=y n1 x 1,...,n l f fr1 x 1,...,r l { v r1 x = 1,...,r l, r r l is odd, w r1 x 1,...,r l, r r l is even. According to the above definitions, the first few lowest order binary Bell polynomials are y x v=v x, y x v, w=w xx + v x, y x,t v, w=w x,t + v x v t, y x v, w=v x +v x w x + v x,. y 4x v, w=w 4x +w x +4v xv x +6v x w x + v 4 x,... Proposition 1 The link between binary Bell polynomials Y n1 x 1,...,n l v, w and the standard Hirota bilinear equation n 1 x 1 n l F Gcanbegivenbytheidentity Y n1 x 1,...,n l v = ln F/G, w = ln FG=FG 1 n 1 x 1 n l F G,. in which n 1 + n + + n l 1, and operators x1 xl are classical Hirota s bilinear operators defined by n 1 x 1 n l F G = x1 x 1 n1 xl x l n l Fx 1,..., G x 1,...,x l x 1 =x 1,...,x l =. IntheparticularcasewhenF= G, equation. becomes G n 1 x 1 n l G G = Y n1 x 1,...,n l 0, q =ln G = { 0, n n l is odd, P n1 x 1,...,n l q, n n l is even,.4 which is also called the case of P-polynomials, P n1 x 1,...,n l q=y n1 x 1,...,n l 0, q =ln G. The first few P-polynomials are, P x q=q x, P x,t q=q xt, P xy q=q xy +q x q xy, P 6x q=q 6x +15q x q 4x +15qx....5

4 Hu and Tao Advances in ifference Equations :1 Page 4 of 1 Equations. and.4 play an important role in connecting bilinear equations with the corresponding NLEEs. Once an NLEE is written in the form of a combination of the Y- polynomials, then its bilinear form can easily be obtained. Proposition The binary Bell polynomials Y n1 x 1,...,n l v, w can be separated into P- polynomials and Y -polynomials, FG 1 n 1 x 1 n l F G = Y n1 x 1,...,n l v, w v=ln F/G,w=ln FG = Y n1 x 1,...,n l v, v + q v=ln F/G,q= ln G n 1 n l l n i = P r1 x 1,...,r l q n 1 + +n l =even r 1 =0 r l =0 i=1 r i Y n1 r 1 x 1,...,n l r l v..6 Under the Hopf-Cole transformation v = ln ψ, that is, ψ = F/G, multi-dimensional binary Bell polynomials Y n1 x 1,...,n l v canbelinearizedintothefollowingform: Y n1 x 1,...,n l v=ln ψ = ψ n1 x 1,...,n l /ψ..7 Equations.6 and.7 provide a direct way to the associated Lax system of NLEEs. Bilinear forms and exact solutions of equation 1.1 Consider the generalized-blmp equation 1.1, setting u = q x,.1 where q is the function of x, y, z, t. Substituting equation.1 intoequation1.1 and integrating with respect to x yields q xy +q xx q xy q yt +q yz =0.. Now according to the P-polynomials.5, equation.can be written as P xy P yt +P yz = 0.. Making a change of the dependent variable, q =ln f u = q x = ln f x, and noting equation.4, we can get the following bilinear form: x y y t + y z f f =0..4 Assume that u = q x y,.5

5 Hu and Tao Advances in ifference Equations :1 Page 5 of 1 where y is an arbitrary differential function of y. Substituting equation.5intoequation 1.1 and integrating with respect to x, the generalized-blmp equation can be written as q xy +q xx q xy +ϕyq xx q yt +q yz =0,.6 where ϕy = y. Equation.6 can be cast into a linear combination form of the P- polynomials, P xy +ϕyp xx P yt +P yz =0..7 Setting q =ln f u = ln f x y, by using equation.4 which connects the Y-polynomialswith the Hirota operators,we can get another bilinear representation of equation 1.1, x y +ϕy x y t + y z f f =0..8 Next, we would like to construct exact solutions for the generalized-blmp equation 1.1 with the aid of the Hirota method. Noting that the bilinear form.8 contains an arbitrary function ϕy= y, we can get some particular solutions of equation 1.1 provided that y is appropriately chosen. So we will compute soliton solutions for equation 1.1 via equation.8. Expand f with respect to a formal parameter ɛ as follows: f =1+f 1 ɛ + f ɛ + f ɛ +,.9 where f i i =1,,... are the functions of x, y, z, andt. Substituting equation.9 into.8 and collecting the coefficients of each order of ɛ yields f 1 xxxy f xxxy f xxxy ϕyf xx f yt +f yz 1 +ϕyf xx f yt +f yz =0,.10a = x y +ϕy x y t + y z f 1 f 1,.10b +ϕyf xx f yt +f yz = x y +ϕy x y t + y z f 1 f,.10c From equations.10a-.10c, we can get f 1 = e y kx k +mz+ mt,.11

6 Hu and Tao Advances in ifference Equations :1 Page 6 of 1 Figure 1 Solitary waves. a Bell-shaped solitary wave with y=sechy, t =0,z =0,k =0.1;b two bell-shaped solitary waves with y=sechy, t =0,z =0,k 1 =0.1,k =0.. and the one-soliton solution for equation 1.1isdenotedby [ u = n 1+e kx y k +mz+ mt] y. Noting that.10a is a linear differential equation, f 1 = e ξ 1 + e ξ, ξ j = k j x y k j + m j z + m jt j =1,.1 is also a solution of equation.10a. Substituting equation.1 intoequation.10b, one can readily obtain f = e ξ 1+ξ +A 1, e A 1 = k 1 k k 1 + k k 1k k 1 + k k 1 + k + k 1k and the two-soliton solution for equation 1.1canbegivenas u = x [ ln 1+e ξ 1 + e ξ + e ξ 1+ξ +A 1 ] y. Two specific solutions of the above one-soliton and two-soliton solutions are plotted in Figures 1 and. 4 Bilinear Bäcklund transformation, associated Lax pair, and infinite conservation laws In this section, we would like to construct the bilinear Bäcklund transformation, the Lax pair, and infinite conservation laws of equation 1.1 via the bilinear form.8.

7 Hu and Tao Advances in ifference Equations :1 Page 7 of 1 Figure Solitary periodic waves. a Solitary periodic wave with y=siny, t =0,z =0,k =0.1;b two solitary periodic waves with y=siny, t =0,z =0,k 1 =0.1,k = Bilinear Bäcklund transformation and associated Lax pair Let q =ln F and q =ln G be two different solutions of equation.6, we have the twofield condition E q Eq= q q xy + q xx q xy q xxq xy +ϕy q q x q q yt + q q yz. 4.1 If we set v = q q / = lnf/g, w = q + q / = lnfg, 4. then equation 4.1can be rewritten as E q Eq / = v xy +w xx v xy + v x w xy +ϕyv xx v yt +v yz. 4. Next, we need to impose a constraint to express equation 4.intheformofy-derivative of Y-polynomials. A simple choice of such a constraint may be Y xy v, w+ϕy=0, 4.4 from which we can get v x v xy + v y v xx + w xxy = Owing to 4.4and4.5, equation 4.canbewrittenas E q Eq / = y Yx v, w Y t v, w+y z v, w =

8 Hu and Tao Advances in ifference Equations :1 Page 8 of 1 Thus, we deduce a coupled system of Y-polynomials Y xy v, w+ϕy=0, Y x v, w Y t v, w+y z v, w= By virtue of equation., one can immediately get the bilinear Bäcklund transformation of equation 1.1, x y + ϕy F G =0, x t + z F G = For obtaining the corresponding Lax pair of system 4.7, one needs to make the Hopf- Cole transformation v = ln ψ. It follows from equations.6and.7that Y t v=ψ t /ψ, Y z v=ψ z /ψ, Y xy v, w=q xy + ψ xy /ψ, Y x v, w=q x ψ x /ψ + ψ x /ψ. 4.9 Then system 4.7islinearizedintoaLaxpair L 1 ψ = ψ xy + q xy + ϕy ψ =0, L ψ = ψ x +q x ψ ψ t +ψ z = It is easy to check that the integrability condition for 4.10 [L 1, L ]ψ = is satisfied if u = q x y is a solution of the generalized-blmp equation Infinite conservation laws To find infinite conservation laws for the generalized-blmp equation 1.1, we would like to introduce a new potential function η = q x q x, it follows from equation 4.that v x = η, w x = q x + η. 4.1 Substituting equation 4.1into4.4yields η y + η 1 x η y + q xy + ϕy= Inserting the expansion η = ε + I n q, q x,...ε n 4.14

9 Hu and Tao Advances in ifference Equations :1 Page 9 of 1 into equation 4.1 and equating the coefficients for power of ε, wethengettherecursion for I n, I 1 = q x + θ 1 x, z, t, I = q x + θ x, z, t, I n = I n 1 n x k=1 y 1 x I k x 1 In 1 k y θ n x, z, t, n ; here θ n x, z, tn =1,,...areundeterminedfunctionsofx, z, t. Rewrite equation 4.6 as the divergence-type form t v y + x v xxy + y vx w xx + v x + z v y = Taking advantage of 4.1 and substituting expansion 4.14 into4.16 leadsto t x 1 In y ε n + x I n xy ε n ] + y [ ε + I n ε q n xx + I x n ε n + ε + I n ε n + z 1 x In y ε n = Comparing the coefficients of ε 1, ε,...yieldsthefollowinginfinitesequenceofconservation laws: F n t + G n x + H y n + J z n, n =1,, In equation 4.18, F n, G n, J n are given by F n = 1 x In y, G n = I n xy, Jn = 1 x In y, n =1,,..., 4.19 and H n are given by H 1 = [ I 1 q xx + I x + I 1 + I ], H = [ I q xx + I x + I 1 I x 1 +I 1 I + I 4], H n = n 1 x + I n q xx + I n+1 n 1 + k=1 n k 1 j=1 k=1 I k I n k x + I n+ + I k I j I n k j, n =,4,... n I k I, n+1 k k=1 4.0

10 Hu and Tao Advances in ifference Equations :1 Page 10 of 1 Moreover, from the coefficients of ε 0 and ε,wecanget I 1 = q xx, I y = I 1 xy, thus we have θ 1 x, z, t=0andθ x, z, t is an arbitrary function of x, z, t.from4.19and 4.0 we readily see that θ n x, z, tn are also arbitrary functions of x, z, t. 5 Another bilinear Bäcklund transformation and traveling wave solutions This section is contributed to construct another bilinear Bäcklund transformation and traveling wave solutions for equation 1.1 based on the bilinear form.4. Assume that f is another solution of the generalized-blmp equation 1.1: x y y t + y z f f = Let us define a function P = [ x y y t + y z f f ] f [ x y y t + y z f f ] f =0. 5. Obviously, if P =0,thenf satisfies equation.4ifandonlyiff satisfies the same equation. Therefore, if we can obtain from P = 0 a system of bilinear equations, B 1 t, x, y, z f f =0, B t, x, y, z f f =0,..., 5. B k t, x, y, z f f =0, where B 1, B,...,B k are undetermined functions, then system 5. provides a bilinear Bäcklund transformation for the generalized-blmp equation 1.1. To this end, we would like to apply the following three exchange formulas for Hirota s bilinear operators: t x a ab t x b ba = x t a b ba, 5.4 t y a ab t y b ba = y t a b ba, 5.5 x ya a b x yb b a [ = x x y a b ba + x a b y b a+6 x y a b x b a ] [ + y x a b ba + x a b x b a ]. 5.6 Equations 5.4and5.5can be found in[11], and equation 5.6is obtained by Wu in[6]. From equations 5.4and5.5, we can get z a a b z b b a = z z a b ba, 5.7 r s a b ba = s r a b ba. 5.8

11 Hu and Tao Advances in ifference Equations :1 Page 11 of 1 Then, by using equations , we can obtain P = [ x yf f f x yf f f ] 4 [ y t f f f y t f f f ] +6 [ y z f f f y z f f f ] = x x y f f ff + x x f f y f f + x 6x y f f x f f + y [ x 8 t +1 z f f ] ff + y x f f x f f. 5.9 By introducing seven new parameters λ i i =1,...,7,equation5.9canbewrittenas P = x x y f f + λ 1 y f f + λ f f ff + x x f f + λ y f f + λ 4 f f y f f + x 6x y f f + λ 5 x f f x f f + y [ x 8 t +1 z λ 1 x + λ 6 f f ] ff + y x f f + λ 7 x f f λ 4 f f x f f = x B1 f f ff + x B f f y f f + x B f f x f f + y B4 f f ff + y B5 f f x f f Taking advantage of equation 5.8and r g g = 0, we immediately see that the coefficients of λ 1, λ,...,λ 7 in equation 5.10 are all zeros. Therefore, we obtain the following bilinear Bäcklund transformation for the generalized-blmp equation 1.1: B 1 f f x y + λ 1 y + λ f f =0, B f f x + λ y + λ 4 f f =0, B f f 6 x y + λ 5 x f f =0, 5.11 B 4 f f x 8 t +1 z λ 1 x + λ 6 f f =0, B 5 f f x + λ 7 x λ 4 f f =0. In the following, we would like to derive explicit solutions the generalized-blmp equation 1.1 by using the bilinear Bäcklund transformation To begin with, we start with a simple solution f =1,fromwhichwecangettheoriginalsolutionu = ln f x =0. Substituting f =1intoequation5.11, one can readily obtain f xxy + λ 1f y + λ f =0, f xx + λ f y + λ 4f =0, 6f xy + λ 5f x =0, f xxx 8f t +1f z λ 1f x + λ 6f =0, f xx + λ 7f x λ 4f =0. Case 1. By setting λ =0, λ 4 =0, λ 6 =0, 5.1 λ 1 = k, λ = k, λ 5 = 6l, λ 7 = k, l

12 Hu and Tao Advances in ifference Equations :1 Page 1 of 1 we get a class of exponential wave solutions of equation 5.1: f =1+εe kx+ly+mz ωt+ξ 0, 5.1 where k, m, ε, ξ 0 are arbitrary constants and l 0,ω = m 1 k.hence,u = ln f x solves the generalized-blmp equation 1.1. Case. Let λ i =0 1 i 7, k, l, m are arbitrary constants and ω = m,itiseasytocheckthat f = kx + ly + mz ωt 5.14 satisfies the bilinear generalized-blmp equation.4, and so u = ln f x = k kx + ly + mz ωt 5.15 gives a class of rational solutions for the generalized-blmp equation Conclusions In this paper, bilinear Bäcklund transformations and explicit solutions of a + 1- dimensional nonlinear equation are investigated. By virtue of the Bell-polynomial approach, two bilinear forms of equation 1.1 are derived and two kinds of bilinear Bäcklund transformations are constructed. Furthermore, explicit solutions for equation 1.1arealso obtained. It is interesting to note that one can obtain different bilinear forms and bilinear Bäcklund transformations via different approaches. We think that there is still much to do to explore more methods of constructing bilinear forms and bilinear Bäcklund transformations for NLEEs. Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally in this article. All authors read and approved the final manuscript. Acknowledgements This work was supported by the Fundamental Research Funds for the Central Universities 015XKMS071. Received: 0 June 016 Accepted: 7 October 016 References 1. Marin, M, Agarwal, RP, Mahmoud, SR: Nonsimple material problems addressed by the Lagrange s identity. Bound. Value Probl. 01, Marin, M: An evolutionary equation in thermoelasticity of dipolar bodies. J. Math. Phys. 40, El-Sayed, AMA, Gaber, M: The Adomian decomposition method for solving partial differential equations of fractal order in finite domains. Phys. Lett. A 59, Golmankhaneh, AK, Khatuni, T, Porghoveh, NA, Baleanu, : Comparison of iterative methods by solving nonlinear Sturm-Liouville, Burgers and Navier-Stokes equations. Cent. Eur. J. Phys. 104, Bhrawy, AH, oha, EH, Baleanu,, Ezz-Eldien, SS: A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations. J. Comput. Phys. 9, Bhrawy, AH, Zaky, MA, Baleanu, : New numerical approximations for space-time fractional burgers equations via a Legendre spectral-collocation method. Rom. Rep. Phys. 67,

13 Hu and Tao Advances in ifference Equations :1 Page 1 of 1 7. Ablowitz, MJ, Clarkson, PA: Soliton, Nonlinear Evolution Equations and Inverse Scatting. Cambridge University Press, New York Matveev, VB, Salle, MA: arboux Transformation and Solitons. Springer, Berlin Gu, CH, Hu, HS, Zhou, ZX: arboux Transformation in Solitons Theory and Geometry Applications. Shangai Science Technology Press, Shanghai Miura, MR: Bäcklund Transformation. Springer, Berlin Rogers, C, Shadwick, WF: Bäcklund Transformations and Their Applications. Academic Press, New York Singh, M: Multi-soliton solutions, bilinear Backlund transformation and Lax pair of nonlinear evolution equation in + 1-dimension. Comput. Methods iffer. Equ., Hirota, R: The irect Method in Soliton Theory. Cambridge University Press, New York Hirota, R: A new form of Bäcklund transformations and its relation to the inverse scattering problem. Prog. Theor. Phys. 5, Belokolos, E, Bobenko, AI, Enolskij, VZ, Its, AR, Matveev, VB: Algebro-Geometric Approach to Nonlinear Integrable Equations. Springer, Berlin Gesztesy, F, Holden, H: Soliton Equations and Their Algebro-Geometric Solutions. Vol. I: imensional Continuous Models. Cambridge University Press, New York Gesztesy, F, Holden, H: Soliton Equations and Their Algebro-Geometric Solutions. Vol. II: imensional iscrete Models. Cambridge University Press, New York Jafarian, A, Ghaderi, P, Golmankhaneh, AK: Construction of soliton solution to the Kadomtsev-Petviashvili-II equation using homotopy analysis method. Rom. Rep. Phys. 65, Gilson, C, Lambert, F, Nimmo, J, Willox, R: On the combinatorics of the Hirota -operators. Proc. R. Soc. Lond. A 45, Lambert, F, Loris, I, Springael, J: Classical arboux transformations and the KP hierarchy. Inverse Probl. 17, Lambert, F, Springael, J: Soliton equations and simple combinatorics. Acta Appl. Math. 10, Fan, EG: The integrability of nonisospectral and variable-coefficient KdV equation with binary Bell polynomials. Phys. Lett. A 75, Hon, YC, Fan, EG: Binary Bell polynomial approach to the non-isospectral and variable-coefficient KP equations. IMA J. Appl. Math. 77, Wang, YH, Chen, Y: Binary Bell polynomial manipulations on the integrability of a generalized + 1-dimensional Korteweg-de Vries equation. J. Math. Anal. Appl. 400, Wang, YH, Chen, Y: Integrability of the modified generalised Vakhnenko equation. J. Math. Phys. 5, Wu, JP: A bilinear Bäcklund transformation and explicit solutions for a + 1-dimensional soliton equation. Chin. Phys. Lett. 5, Ma, WX, Abdeljabbar, A: A bilinear Bäcklund transformation of a + 1-dimensional generalized KP equation. Appl. Math. Lett. 5, Gilson, CR, Nimmo, JJC, Willox, R: A + 1-dimensional generalization of the AKNS shallow-water wave-equation. Phys. Lett. A 180, Estevez, PG, Leble, S: A wave-equation in + 1: Painlevé analysis and solutions. Inverse Probl. 11, Tang, XY: What will happen when a dromion meets with a ghoston? Phys. Lett. A 14, Luo, L: New exact solutions and Bäcklund transformation for Boiti-Leon-Manna-Pempinelli equation. Phys. Lett. A 75, Luo, L, Gui, SH: Bäcklund transformation and exact solutions of a + 1-dimensional nonlinear equation. Chin. Ann. Math., Ser. A,