Integrability for two types of (2+1)-dimensional generalized Sharma-Tasso-Olver integro-differential equations

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1 MM Research Preprints, MMRC, AMSS, Academia Sinica, Beijing No. 22, December 2003 Integrability for two types of (2+1)-dimensional generalized Sharma-Tasso-Olver integro-differential equations Zhenya Yan Key Laboratory of Mathematics Mechanization, Institute of systems Science, AMSS, Chinese Academy of Sciences, Beijing , P.R. China Abstract. In this paper, two hierarchies of (2+1)-dimensional generalized Burgers integro-differential equations are derived by using two types of Cole-Hopf transformations, u(x, y, t) = φ y (x, y, t)/φ(x, y, t) and u(x, y, t) = φ x (x, y, t)/φ(x, y, t), in (2+1) dimensions and two recursion formulae. As a reduction, two representative systems of the (2+1)-dimensional generalized Sharma-Tasso-Olver (GSTO) integro-differential equations, [ simply called the GSTO-I : u t = u xxy + (uu x ) x + (u 2 y 1 u x ) x + (u y 1 u x ) xy, and the GSTO-II : u t = u xxy + (uu y ) x + (u 2 x 1 u y ) x + (u x 1 u y ) xx ], are given, respectively. It is proved that these two GSTO equations both have the Painlevé property such that Bäcklund transformations are gained to construct many types of new exact solutions including multi-soliton-like solutions, shock-like wave solutions, infinitely many rational solutions. The single soliton solutions obtained are used to show that the variables, u x for GSTO-I and u y for GSTO-II, admit exponentially localized solutions (dromion solutions) rather than the physical field u(x, y, t) itself. Moreover, we also prove that the GSTO-II equation is C-integrable and possesses bi-hamiltonian structure. Keywords:Cole-Hopf transformation; Painlevé analysis; Bäcklund transformation; C- integrability; Exact solution; Bi-Hamiltonian structure PACS: Ik; Yv; Ge 1. Introduction It is of important significance in soliton theory and integrable systems to seek more (2+1)- dimensional or even higher-dimensional nonlinear evolution equations and to further study their integrable properties such as bi-hamiltonian structure, conservation laws, symmetry, Painlevé integrability, C-integrability, S-integrability, multi-soliton solutions, etc[1-4]. To date, there exist many powerful techniques to study them such as inverse scattering method, Painlevé test method, Hirota s bilinear method, Bäckulnd transformation, Darboux transformation and Lie group method, etc. Among them, the Painlevé analysis[3,4] is considered to be a more powerful and systematic method to identify the integrability of nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) to construct many integrable properties in a systematic approach. Singularity analysis admitting the Painlevé property was presented by Ablowitz et al.[3] for ODEs and then extended by Weiss, Tabor, and Carnevale (WTC)[4] to PDEs. And then Kruskal[1] assumed the manifold ansatz

2 Integrability 303 φ(x, y, t) in the WTC method by using a simple form φ(x, y, t) = x+ψ(y, t). Many important nonlinear evolution equations were proved to have the Painlevé properties[1,4]. Extending linearization achieved through the Cole-Hopf transformation to equations containing as highest derivatives odd space derivatives was used to find new odd members of the Burgers hierarchy by Tasso and Sharma[5,9]. The most prominent example (called Sharma-Tasso-Olver(STO) equation) is given by[5-9] u t = (u 3 ) x (u2 ) xx + u xxx, (1) But in fact the physical field u = u(x, t) which describes the wave motion should exist in (2+1)-dimensional or even higher-dimensional space. As far as we know, (2+1)-dimensional or even higher-dimensional extensions of (1) were not reported before. Motivated by the Tasso s idea and the assumption of higher-dimensional extensions, we would like to construct two hierarchies of generalized Burgers equations in (2+1) dimensions whose representative systems are GSTO-I equation : and GSTO-II equation : u t = u xxy + (uu x ) x + (u 2 y 1 u x ) x + (u y 1 u x ) xy, (2) u t = u xxy + (uu y ) x + (u 2 x 1 u y ) x + (u x 1 u y ) xx. (3) respectively, by means of two types of Cole-Hopf transformations u(x, y, t) = φ y(x, y, t) φ(x, y, t), u(x, y, t) = φ x(x, y, t) φ(x, y, t), (4) in (2+1)-dimensional spaces, where y 1 = 1 2 ( y y ), y y 1 = y 1 y = 1, x 1 = 1 2 ( x x ), x x 1 = x 1 x = 1. (2) and (3) can be understood as (2+1)-dimensional extensions of (1), because in the one-dimensional reduction u = u(x + y, t), (2) and (3) both reduce to (1). This is why we refer to (2) and (3) as the (2+1)-dimensional GSTO integrodifferential equations. The motions described by (2) and (3) are isolated waves, located in a small part of spaces. These will be illustrated below. Since the concept of dromions of Davey-Stewartson equation was firstly presented by Boiti et al.[10], many nonlinear soliton equations in (2+1) dimensions have been shown to possess the dromions such as the generalized KdV equation, the Kadomtsev-Petviashvilli (KP) equation, the dispersive long wave equation and Nizhnik-Novikov-Veselov equation, etc.[10-18]. In addition, with the aid of Backlund transformation and some generalized ansatz, some abundant soliton-like solutions and infinitely many rational solutions as well as other types of solutions were derived for many (2+1)-dimensional and (3+1)-dimensional nonlinear evolution equations[17-28]. A natural problem is whether (2) and (3) also possess the dromion solutions and other types of soliton-like solutions and infinitely many rational solutions. The rest of the paper is organized as follows: In Section 2, we derive the two whole hierarchies including (2) and (3) by using two types of Cole-Hopf transformations (4). In Section 3, it is shown that equation (2) has Painlevé property in terms of WTC method. In Section 4,

3 304 Zhenya Yan a Bäcklund transformations of (2) is obtained to seek some new multi-soliton-like solutions, multi-soliton solutions and infinitely many rational solutions, which show that the variable u x admits exponentially localized solutions rather than the physical field u(x, y, t) itself. In Section 5, it is prove that equation (3) also has the Painlevé property and is C-integrable. Moreover we construct its many multi-soliton-like solutions, multi-soliton solutions and infinitely many rational solutions. In Section 6. we give the bi-hamiltonian structure of (3). Some conclusions and problems are given in Section Two hierarchy of (2+1)-dimensional generalized Burgers equations 2.1 The first hierarchy of generalized Burgers equations We suppose that the physical filed u = u(x, y, t) in a certain nonlinear evolution equation has the solution in the form u(x, y, t) = y ln φ(x, y, t) = φ y(x, y, t) φ(x, y, t), (5) which is a Cole-Hopf transformation in (2+1) dimensions. It is clear to see that from (5) [ ( 1 y ) ] φ(x, y, t) = exp( y 1 u) = exp u(x, y, t)dy, (6) 2 y where t is a temporal variable, and x, y are spatial variables (hereafter), y 1 = 1 2 ( y y ), y 1 y 1 y = 1, Generally speaking, the temporal derivative of φ(x, y, t) can be chosen as the linear form of φ, for example φ t (x, y, t) = [ y 1 V nx (u, u x, u y,...)]φ(x, y, t). (7) where V n (u, u x, u y,...) s are unknown function of u and its spatial derivatives. It is easy to see that (5) and (7) compose a generalized Lax pair of the equation which u satisfies. The compatibility condition φ yt = φ ty of (5) and (7) is zero curvature equation that is to say u t V nx + [u, 1 y V nx ] = 0, (8) u t = V nx (u, u x, u y,...) = x V n(u, u x, u y,...). (9) In the following we would like to construct a sequence of functions V n (u, u x, u y,...) y = V n+1 = uv n + y V n, V 0 = 1 y u x, n = 0, 1, 2,... (10) which is different from the inducing formula[9,29]. Therefore the first hierarchy of generalized Burgers equations in (2+1) dimensions are defined by (9) and (10). By using the recursion operator Ψ n = x (u + y ) n, the hierarchy can be rewritten as u t = V nx = Ψ n ( y 1 u x ) = [ x (u + y ) n ]( y 1 u x ), n = 1, 2,... (9 )

4 Integrability 305 Remark 1: (i) When n = 1, we have the (2+1)-dimensional generalized Burgers equation u t = u xx + u x y 1 u x + u y 1 u xx, (11a) which has been proved to possess Painleve test and Backlund transformation as well as many types of exact solutions[19]. (ii) When n = 2, we have the (2+1)-dimensional GSTO equation (2) which will be studied in the following. (iii) When n > 2, we can obtain more higher-order nonlinear evolution equations. For example, for n = 3, we have the forth-order nonlinear evolution integro-differential equation u t = u xxyy + (uu x ) xy + (uu xy ) x (u3 ) xx + (u 3 y 1 u x ) x (u2 y 1 u x ) xy + (u y 1 u x ) xyy. (11b) Remark 2. The mixed equations may also be established by changing V n into the form N a j V j, where V j s are defined by (10) and a js constants. For example, when N = 2, we can obtain the first type of (2+1)-dimensional generalized STO-Burgers equation u t = a 1 u xx + a 1 (u 1 y u x ) x + a 2 u xxy + a 2 (uu x ) x + a 2 (u 2 y 1 u x ) x + a 2 (u y 1 u x ) xy. (11c) Remark 3. When u = u(x + y, t), (9 ) reduces to the (1+1)-dimensional hierarchy of Burgers equations due to Tasso and Sharma[5,9]. When u = 1 2u(x + y, t), (9 ) reduces to the flow of Burgers equations due to Olver[6]. 2.2 The second hierarchy of generalized Burgers equations Similar to the section 2.1, we suppose that another physical filed u = u(x, y, t) in a certain nonlinear evolution equation has the solution in the form u(x, y, t) = x ln φ(x, y, t) = φ x(x, y, t) φ(x, y, t), (12) which is also a Cole-Hopf transformation in (2+1) dimensions. It is clear to see that from (12) [ ( 1 x ) ] φ(x, y, t) = exp( x 1 u) = exp u(x, y, t)dx (13) 2 x where x 1 = 1 2 ( x x ), x x 1 = x 1 x = 1. In general, the temporal derivative of φ(x, y, t) can be chosen as the linear form of φ φ t (x, y, t) = V n (u, u x, u y,...)φ(x, y, t). (14) where V n (u, u x, u y,...) s are unknown function of u and its spatial derivatives. It is easy to see that (12) and (14) lead to a generalized Lax pair of the equation which u satisfies. The compatibility condition φ xt = φ tx of (12) and (14) is zero curvature equation u t V nx + [u, V nx ] = 0, (15)

5 306 Zhenya Yan that is u t = V nx (u, u y, u x...) = x V n(u, u x, u y,...), n = 0, 1, 2,... (16) In the following we construct a sequence of functions V n (u, u x, u y,...) V n+1 = uv n + x V n, V 0 = 1 x u y, (17) which is also different from the inducing formula[9,29]. Therefore, the second hierarchy of generalized Burgers integro-differential equations in (2+1) dimensions are obtained by (16) and (17). The hierarchy can be rewritten as u t = V nx = Ψ n ( x 1 u y ) = [ x (u + x ) n ]( x 1 u y ), n = 1, 2,... (16 ) by using the recursion operator Ψ n = x (u + x ) n. Remark 4. (a) When n = 1, we from (16) and (17) get the (2+1)-dimensional generalized Burgers equation u t = u xy + uu y + u x 1 x u y, (18) which had been proved to possess the Painleve property and auto-backlund transformation, as well as to be C-integrable. Moreover many types of exact solutions were obtained of (18)[20,21]. Recently, we have given many types of (1+1)-dimensional symmetry reductions and (1+1)-dimensional condition symmetry reductions 0f (18) [30]. (b) When n = 2, we have the (2+1)-dimensional GSTO equation (3) which will be investigated below. (c) When n > 2, we can obtain more higher-order nonlinear evolution equations. For example, for n = 3, we have the forth-order nonlinear evolution integro-differential equation u t = u xxxy + (uu y ) xx + (uu xy ) x (u3 ) xy + (u 3 x 1 u y ) x (u2 x 1 u y ) xx + (u x 1 u y ) xxx. (19a) Remark 5. The mixed equations may also be established by changing V n into the form M b j V j, where V j s are defined by (17) and b js constants. For example, when N = 2, we can obtain the second type of (2+1)-dimensional generalized STO-Burgers equation u t = b 1 u xy + b 1 (u 1 x u y ) x + b 2 u xxy + b 2 (uu y ) x + b 2 (u 2 x 1 u y ) x + b 2 (u x 1 u y ) xx. (19b) Remark 6. When u = u(x + y, t), (17 ) reduces to the (1+1)-dimensional hierarchy of Burgers equations due to Tasso and Sharma[5,9]. When u = 1 2u(x + y, t), (17 ) reduces to the flow of Burgers equations due to Olver[6]. 3. The Painlevé analysis of the GSTO-I equation (2) To investigate the singularity structure aspects of (2) by using the WTC method[4], we shall first introduce the transformation u x = w y and convert (2) into a coupled nonlinear evolution partial differential equations u t = u xxy + (u 2 w) x + (uw y ) x + (uw) xy, (20a)

6 Integrability 307 u x = w y, (20b) In the coupled equations, u(x, y, t) denotes the physical field and w(x, y, t) its potential. In order to carry out a singularity structure analysis, we effect a local Laurent expansion in the neighborhood of a non-characteristic singular manifold, φ(x, y, t) = 0(φ x φ y 0). Assuming the leading orders of the solutions of (20) to have the form u = u 0 φ α, w = w 0 φ β, (21) where u 0 and w 0 are two analytic functions of x, y, t, and α and β are integers to be determined later. Substituting (21) into (20) and balancing the nonlinear terms against the dominant linear terms give rise to α = β = 1, u 0 = φ y, w 0 = φ x, (22) In the following we consider the Laurent series expansion of the solutions in the neighborhood of the singular manifold u = u j φ j 1 = φ y φ + u j φ j 1, w = w j φ j 1 = φ x φ + w j φ j 1. (23) j=0 j=0 Substituting (23) into (20) yields u j 3,t (j 3)u j 2 φ t + (j 1)(j 2)(j 3)u j φ 2 xφ y + 2(j 2)(j 3)u j 1 φ x φ xy +(j 2)(j 3)u j 1 φ xx φ y + (j 2)(j 3)u j 1,y φ 2 x + 2(j 2)(j 3)u j 1,x φ x φ y +2u j 2,xy φ x + (j 3)u j 2,xx φ y + 2(j 3)u j 2,x φ xy + (j 3)u j 2,y φ xx + (j 3)u j 2 φ xxy j i +u j 3,xxy + {2w j i u i k [u k 1,x + (k 1)φ x u k ] + u j i u i k [u k 1,x + (k 1)φ x u k ]} i=0 k=0 j + {w j i [u i 2,xy + (i 2)u i 1,x φ y + (i 2)u i 1,y φ x + (i 2)u i 1 φ xy + (i 1)(i 2)u i φ x φ y ] i=0 +u j i,y [w i 2,x + (i 2)w i 1 φ x ] + (j i 1)u j i φ y [w i 1,x + (i 1)w i φ x ] + 2u j i [w i 2,xy +(i 2)w i 1,x φ y + (i 2)w i 1,y φ x + (i 2)w i 1 φ xy + (i 1)(i 2)w i φ x φ y ] +2w j i,y [u i 2,x + (i 2)u i 1 φ x ] + 2(j i 1)w j i φ y [u i 1,x + (i 1)u i φ x ]} = 0 (24a) u j 1,x + (j 1)u j φ x w j 1,y (j 1)w j φ y = 0. (24b) the resonance, that is, powers at which arbitrary function enter into the series (23), can be evaluated by comparing the coefficients of (φ j 4, φ j 2 ) of (24a) and (24b), we have ( (j 1) 2 (j 3)φ 2 xφ y 2(j 1))(j 3)φ x φ 2 ) ( ) y uj = 0. (25) (j 1)φ x (j 1)φ y w j Therefore we can arrive at the resonances from (25) j = 1, 1, 1, 3. (26)

7 308 Zhenya Yan The resonance at j = 1 represents the arbitrariness of the singularity manifold φ(x, y, t) = 0. In the following let us examine the cases j = 1, 2 and 3, successively. Case A. When j = 1, system (24) reduces to (2u 2 0φ x 2u 0 φ x φ y )w 1 + (4u 0 w 0 φ x +4w 0 φ x φ y )u 1 = 4u 0x φ x φ y + 2u 0 φ x φ y + 2u 0 (φ x φ y ) x + 2u 0y φ 2 x + 2w 0 u 0 u 0x + u 2 0w 0x 3w 0 u 0x φ y 2w 0 u 0y φ x 3u 0 w 0 φ xy 3u 0 w 0x φ y 4u 0 w yx φ x, u 0x = u 0y. (27a) (27b) Substituting (22) into system (27), we can easily verify that system (27) identically vanishes. Case B. When j = 2, system (24) reduces to ( φ 2 x φ y φ x φ 2 ) ( ) y u2 = φ x φ y w 2 ( F1 F 2 ), (28) where F 1 = φ x φ xyy 2φ 2 xy φ y φ t + 2φ xx φ yy 2w w u 1 φ xφ y w 2 1φ 2 x It is easy to see that (28) gives rise to w 1 φ x φ yy u 1y φ 2 x 2w 1y φ x φ y 2u 1 φ x φ xy, F 2 = w 1y u 1x. (29a) (29b) u 2 = F 1 + φ x φ y F 2 2φ 2, w 2 = F 1 φ x φ y F 2 xφ y 2φ x φ 2. (30) y Case C. When j = 3, system (24) reduces to u 0t + u 0xxy + 2(w 1 u 1 φ xy + u 1 u 1x φ x + w 1 u 1x φ y + w 2 φ x φ xy ) + u 2 1φ xx + 2u 1 w 1x φ y +w 1 φ xyy + u 1xy φ x + 3u 2x φ x φ y w 2 φ y φ xy + u 2 φ x φ xy w 2 φ y φ xy + w 1x φ yy +u 1y φ xx + u 2 φ y φ xx w 1x φ 2 y + 2(u 1 φ xxy + w 1xy φ y + w 2x φ 2 y u 2 φ y φ xx +2w 2 φ y φ xy + u 1x φ xy + w 1y φ xy u 1x φ x φ y ) = 0, (31a) 2u 3 φ x 2w 3 φ y = w 2y u 2x. (31b) Because (31a) is independent of u 3 and w 3, but consistent with the above results, we are left with only the (31b) for two variables u 3 or w 3 and so again one of them must be arbitrary. Therefore the general solution (u(x, y, t), w(x, y, t)) of (20) admits the required number of arbitrary functions, without the introduction of any movable critical manifold, thereby satisfying the Painleve property. Thus the equation (2) passes the Painlevé test. 4. Bäcklund transformation and exact solutions of (2) 4.1. Bäcklund transformation

8 Integrability 309 To deduce a Bäcklund transformation of (2), we truncate the Laurent series at the constant level term O(φ 0 ), that is to say, u j = w j = 0 for j 2, to give rises to u(x, y, t) = u 0 φ 1 + u 1 = y ln φ + u 1, u y = w x. (32) if we assume that u 1 is a solution of (2), then (32) makes (2) reduce to φ[ φ ty + φ xxyy + 2u 1 φ xxy + y 1 u 1x φ xyy + (u 1y + u 2 1)φ xx +(4u 1x + 2u 1 1 y u 1x )φ xy + y 1 u 1xx φ yy + (u 1xy + 2u 1 u 1x )φ x +2(u 1xx + (u 1 y 1 u 1x ) x )φ y ] + φ t φ y φ x [φ xyy + 2u 1x φ y + 2u 1 φ xy +(u 1y + u 2 1)φ x + 2u 1 y 1 u 1x φ y + y 1 u 1x φ yy ] = 0, (33) Therefore (32) and (34) compose an auto-backlund transformation of (2). For given a solution u 1 of (2), if we can determine the trivial function φ from (33), then another solution of (2) will be gotten by means of (32). If we set u 1 = 0, then (32) reduces to the Cole-Hopf transformation under which, (2) can be bi-linearized as from (33) u(x, y, t) = u 0 φ 1 + u 1 = y ln φ, (34) φ t φ y φ x φ xyy φφ ty + φφ xxyy = 0. (35) Proposition 1. The (2+1)-dimensional GSTO-I equation (2) is Painleve integrable and possesses a Backlund transformation Soliton-like solutions with u 1 (x, y, t) = u 1 (y) To seek more types of one soliton-like solutions of (3) by means of the Backlund transforamtion obtained, we assume that the seek solution of (2) is u 1 (x, y, t) = u 1 (y) and the function φ is of the form φ(x, y, t) = P (y, t) + exp[θ(y, t)x + Ψ(y, t)], (36) where P (y, t) 0, Θ(y, t) and Ψ(y, t) are differentiable functions of y and t only to be determined. With the aid of symbolic computation,maple, we substitute (36) into (33) and set to zero the coefficients of these terms x j e k(θx+ψ) to lead to an over-determine system of nonlinear partial differential equations Θ y (Θ t Θ 2 Θ y ) = 0, Θ ty 2ΘΘ 2 y = 0, P t P y P P ty = 0, Ψ ty + 2Θ 2 y + ΘΘ yy + 2ΘΘ y Ψ y + 2u 1 ΘΘ y = 0, P (4ΘΘ 2 y Θ t Ψ y Θ y Ψ t Θ ty + 2Θ 2 Θ y Ψ y + 2u 1 Θ 2 Θ y + Θ 2 Θ yy ) +P y Θ t + P t Θ y + = 0, P y Ψ t + P t Ψ y P ty + P [ Ψ ty Ψ t Ψ y + 2(ΘΘ y ) y + 4ΘΘ y Ψ y + Θ 2 Ψ yy +Θ 2 Ψ 2 y + 4u 1 ΘΘ y + 2u 1 Θ 2 Ψ y + (u u 1y)Θ 2 ] = 0. (37)

9 310 Zhenya Yan By simplifying (37) we get the equivalent system as Θ(y, t) = θ(t) 0, P (y, t) = p(y)q(t) 0, Ψ(y, t) = F (y) + G(t), θ (t)[p (y) p(y)f (y)] = 0, (38) p (y)q(t)g (t) + p(y)q (t)f (y) p (y)q (t) + p(y)q(t)[ G (t)f (y) +θ 2 (t)f (y) + θ 2 (t)f 2 (y) + 2u 1 θ 2 (t)f (y) + (u u 1y)θ 2 (t)] = 0. where the prime denotes the derivative, θ(t), p(y), q(t), F (y) and G(t) are smooth functions of indicated variables. Therefore we get the shock-like wave solution of (2) u = 1 [ ] p (y) 2 p(y) + F (y) tanh 1 [θ(t)x + F (y) + G(t) ln[p(y)q(t)]] [ p ] (y) 2 p(y) + F (y) + u 1 (y), p(y)q(t) > 0. (39a) and singular soliton-like solution solution u = 1 [ ] p (y) 2 p(y) + F (y) coth 1 [θ(t)x + F (y) + G(t) ln[p(y)q(t)]] [ p ] (y) 2 p(y) + F (y) + u 1 (y), p(y)q(t) < 0. (39b) In the following we only discuss this type of solution (39a). The other singular solution (39b) also has the similar cases. From the fourth equation of (38), there exist two possible cases to be discussed below : Case When θ (t) 0, we have The substitution of (40) into the last equation in (38) yields p (y) = F (y)p(y). (40) F (y) + F 2 (y) + 2u 1 F (y) + u u 1y = 0. (41) Making the transformation F (y) = f(y), we reduce (41) to the famous Riccati equation which just has the general solution f (y) + f 2 (y) + 2u 1 f(y) + u u 1y = 0. (42) f(y) = 1 y y 0 u 1 (y). (43)

10 Integrability 311 Therefore we have from (40) and (43) y ( y ) F (y) = ln y y 0 u1 (y )dy, p(y) = C(y y 0 ) exp u1 (y )dy. (44) Therefore we have the shock-like wave solution of (2) from (39a) and (44) [ ] 1 u = u 1 (y) tanh 1 y y 0 2 [θ(t)x + G(t) ln(cq(t))] + u 1(y), Cq(t) > 0. (45) where θ(t), G(t), q(t) are all arbitrary functions of t only, and C, y 0 are constants. Case When θ (t) = 0, i.e., θ(t) = θ = const 0, we have from (38) Θ(y, t) = θ(t) = θ = const 0, P (y, t) = p(y)q(t) 0, Ψ(y, t) = F (y) + G(t), (46) p (y)q(t)g (t) + p(y)q (t)f (y) p (y)q (t) + p(y)q(t)[ G (t)f (y) +θ 2 F (y) + θ 2 F 2 (y) + 2u 1 θ 2 F (y) + (u u 1y)θ 2 ] = 0. In the following we give two special subcases: Case 4.2.2a When q (t) = q(t)(g (t) λ), where λ is a constant, we have from (46) F (y) + (2u 1 (y) λ ) θ 2 F (y) + F 2 (y) + u 1y (y) + u 2 1(y) + λp (y) θ 2 = 0, (47) p(y) Making the transformation F (y) = f(y), we reduce (47) to the famous Riccati equation for f(y) f (y) + (2u 1 (y) λ ) θ 2 f(y) + f 2 (y) + u 1y (y) + u 2 1(y) + λp (y) θ 2 = 0, (48) p(y) from which we can gain the function p(y) as p(y) = exp { θ2 λ [f(y) + u 1(y)] θ2 λ where f(y) is an arbitrary function of y only. Therefore we have the shock-like wave solution of (2) tanh 1 2 [ y [(2u 1 (s) λ ) ] } θ 2 f(s) + f 2 (s) + u 2 1(s) ds. (49) u = θ2 2λ [f (y) + u 1(y) + (f(y) + u 1 (y)) 2 ] ] y[f(y θx + λt + θ2 λ [f(y) + u 1(y)] + θ2 ) + u 1 (y )] 2 dy λ

11 312 Zhenya Yan θ2 2λ [f (y) + u 1(y) + (f(y) + u 1 (y)) 2 ] + f(y) + u 1 (y). (50) In particular, when p(y) = c 3 = const., u 1 (y) = 1 y, we from (48) get f(y) = where C is an integration constant. Thus we have the special shock-like wave solution of (2) u = λ + λy 2 exp( λ θ 2 y) λcy 2 λy + θ 2 y 2 exp( λ θ 2 y). (51) λ + λy 2 exp( λ y) θ 2 2[λCy 2 λy + θ 2 y 2 exp( λ y)] [1 + tanh 1 y λ + λy 2 2 ( exp( λ y θ ) 2 λcy θ 2 λy + θ 2 y 2 exp( λ y 2 θ ) dy 2 +θx + λt ln c 3 )] + 1 y, c 3 > 0. (52) 4.3. Soliton-like solutions with u 1 (x, y, t) = u 1 = const In the following we will seek another type of one soliton-like solutions of (2). By inspection, we assume that the seek solution of (2) is u 1 (x, y, t) = u 1 = const. and the unknown function φ is of the y-linear formal solution φ(x, y, t) = Ω(x, t) + exp[γ(x, t)y + Σ(x, t)], (56) where Ω(x, t), Γ(x, t), Σ(x, t) are differentiable functions of x and t only to be determined. With the aid of symbolic computation,maple, we substitute (56) into (33) and set to zero the coefficients of these terms y j e k(γy+σ) to lead to Γ 2 x(γ + u 1 ) 2 = 0, (Γ + u 1 ) 2 Σ xx Γ t = 0, u 2 1 (ΩΩ xx Ω 2 x) ΩΓ t = 0, (57) Ω t Γ ΩΓΣ t ΩΓΓ t Ω x Σ x (Γ 2 + 2u 1 Γ + 2u 2 1 ) +ΩΣ 2 x(γ + u 1 ) 2 + u 2 1 Ω xx = 0, Therefore we get the shock-like wave solutions of (2) with Ω(x, t) > 0 u = 1 2 Γ(x, t) tanh 1 2 [Γ(x, t)y + Σ(x, t) ln Ω(x, t)] Γ(x, t) + u 1, (58a) and singular soliton-like solutions of (2) with Ω(x, t) < 0 u = 1 2 Γ(x, t) coth 1 2 [Γ(x, t)y + Σ(x, t) ln Ω(x, t) ] Γ(x, t) + u 1, (58b)

12 Integrability 313 where Ω(x, t), Γ(x, t), Σ(x, t) are defined by (57) In the following we will give some special solutions of (2) from (57) and (58a). Case When Γ = γ = const. 0, u 1 = 0, we have the solutions from (57) Σ(x, t) = σ(t)x + ρ(t), Ω(x, t) = F ( t ) x + γ σ(t )dt exp[σ(t)x + ρ(t)]. (59) ( where σ(t), ρ(t) and F x + γ t σ(t )dt ) are all arbitrary smooth functions of their indicated variables. Therefore we have the solution of (2) u = 1 2 γ tanh 1 2 [ ( t )] γy ln F x + γ σ(t )dt + 1 γ, (60) 2 Case When Γ = const = γ = u 1 0, (57) reduces to ΩΩ xx Ω 2 x, Ω t Γ ΩΓΣ t Ω x Σ x Γ 2 + Γ 2 Ω xx = 0. from which we have the general solutions t ( t ) Σ(x, t) = ln f(t) + γg(t )dt + G x γg(t )dt, (61) Ω(x, t) = f(t) exp(g(t)x). (62) where f(t), g(t) and G(x t γg(t )dt ) are all arbitrary smooth functions of their indicated variables. Therefore we have the solution of (2) u = 1 2 γ tanh 1 2 [ t ( t ) ] γy + γg(t )dt + G x γg(t )dt g(t)x 1 γ, (63) Multi-soliton solutions To construct exact soliton solutions of (2), the simplest way is to solve its bilinear form (35) by using a power series such as φ(x, y, t) = 1 + ɛφ (1) + ɛ 2 φ (2) + ɛ 3 φ (3) +, (64) where ɛ is a small parameter. Substituting (64) into (35) and comparing the coefficients of various powers of ɛ, we find the following sets of linear equations ɛ 2 : φ (2) ty ɛ : φ (1) ty φ(1) xxyy = 0 (65) φ(2) xxyy = φ (1) t φ (1) y φ (1) x φ (1) xyy φ (1) (φ (1) ty φ(1) xxyy). (66)

13 314 Zhenya Yan etc. Therefore (65) gives rise to N φ (1) = exp[k i x + f i (l i y ki 2 l i t)] + g(x, t) (67) i=1 where f i and g are arbitrary functions of l i y k 2 i l it and x, t, respectively; k i, l i are arbitrary constants. Substituting (67) into (66) etc, the solution for φ (j) (j 2) can be obtained. Case One-soliton solution To construct one-soliton solution, N = 1, and we take Therefore we have the shock wave solution of (2) φ (1) = exp(k 1 x + l 1 y + k 2 1l 1 t + ξ 1 ), (68) u = l 1 2 [tanh 1 2 (k 1x + l 1 y + k 2 1l 1 t + ξ 1 ) + 1] (69) It is clearly sen that the field u is not exponentially localized in all directions. However for the two potentials v 1 = u x and v 2 = u y, that is v 1 = u x = 1 4 k 1l 1 sech (k 1x + l 1 y + k 2 1l 1 t + ξ 1 ) (70) v 2 = u y = 1 4 l2 1sech (k 1x + l 1 y + k 2 1l 1 t + ξ 1 ) (71) Remark 7. It is easy to know that when k 1 0 and l 1 0, v 1 0. Therefore the solution v 1 = u x is dromion solution. Remark 8. It is clear to see that the solution v 2 is finite on the camber k 1 x + l 1 y + k 2 1 l 1t + ξ 1 = 0 and decays exponentially away for the camber. Therefore we call this type of solution the camber soliton or camber solitary wave[15]. Case Multi-soliton solutions We can obtain the solution of (35) in the form N φ(x, y, t) = c 0 + c i exp(k i x + k i ly + ki 3 lt + ξ i ), (72) i=1 under (34), we have the multi-soliton solutions of (2) u = where k i, l, c i, ξ i are all constants Infinitely many rational solutions Ni=1 c i k i l exp(k i x + k i ly + k 3 i lt + ξ i) c 0 + N i=1 c i exp(k i x + k i ly + k 3 i lt + ξ i). (73)

14 Integrability 315 We assume that (35) has the solution n φ(x, y, t) = i (x + ly)(t t 0 ) i = 0 (x + ly) + 1 (x + ly)(t t 0 ) i=0 + + n (x + ly)(t t 0 ) n. (74) where l, t 0 are constants, and is functions of x + ly to be determined. Substituting (52) into (35) and comparing the coefficients of various power of t, we have (t t 0 ) n : n (x + ly) = 0, (t t 0 ) n 1 : n 1 (x + ly) = nl 1 n, (75) (t t 0 ) : 1 (x + ly) = 2l 1 2, (t t 0 ) 0 : 0 (x + ly) = l 1 1. where the prime denotes the derivatives. From (75), we have the recursion formula ( ) 3(n+1 i) i (x + ly) = l i n n (n i)! n i a j (x + ly) 3(n+1 i) j, i = 0, 1, 2, 3,... (76) (3(n + 1 i) j)! where a js are constants. Therefore we have the infinitely many rational solutions of (2) from (34) [ ( ) ] ni=0 n l 1+i n 3(n i)+2 a (n i)! j (x+ly) 3(n i) j+2 n i (3(n i) j+2)! (t t 0 ) i u = [ ( ) ]. (77) ni=0 n 3(n+1 i) l i n a (n i)! j (x+ly) 3(n+1 i) j n i (3(n+1 i) j)! (t t 0 ) i In particular, when n = 1, we give the rational solution of (2) u = {l[2a 1 (x + ly) + a 2 ](t t 0 ) a 1(x + ly) a 2(x + ly) a 3(x + ly) 2 +a 4 (x + ly) + a 5 }{[a 1 (x + ly) 2 + a 2 (x + ly) + a 3 ](t t 0 ) + l 1 [ 1 60 a 1(x + ly) a 2(x + ly) a 3(x + ly) a 4(x + ly) 2 + a 5 (x + ly) + a 6 ]} 1. (78) where t 0, a is(i = 1, 2,..., 6) are constants The mixed solutions

15 316 Zhenya Yan In addition, itis clear to see that (35) has the formal solutions φ(x, y, t) = m l i m (m i)! i=0 where A i, B i, k i, l are constants. Thus we give the mixed solutions of (2) u = [ ( mi=0 m l 1+i m (m i)! mi=0 [ l i m (m i)! where ξ i = k i x + k i ly + k 3 i lt. ( m ) 3(m+1 i) m i A j (x + ly) 3(m+1 i) j t i (3(m + 1 i) j)! n + B i exp(k i x + k i ly + ki 3 lt). (79) i=1 m i ( m m i ) 3(m i)+2 ) 3(m+1 i) ] A j (x+ly) 3(m i) j+2 (3(m i) j+2)! A j (x+ly) 3(m+1 i) j (3(m+1 i) j)! t i + n i=1 B i k i le ξ i ] t i + n i=1 B i e ξ i. (80) 5. Painlevé analysis, Bäcklund transformation and exact solutions of equation (3) 5.1. Painlevé analysis Similar to section 3, to investigate the singularity structure aspects of (3), we shall first introduce the transformation u y = v x and convert (3) into a coupled nonlinear partial differential equations u t = u xxy + (u 2 v) x + (uv x ) x + (uv) xx, (81a) u y = v x, (81b) In the coupled equation, u(x, y, t) denotes the physical field and v(x, y, t) some potential. In order to carry out a singularity structure analysis, we effect a local Laurent expansion in the neighborhood of a non-characteristic singular manifold, φ(x, y, t) = 0, (φ x φ y 0). Assuming the leading orders of the solutions of (81) to have the form u = u 0 φ α, v = v 0 φ β, (82) where u 0 and v 0 are two analytic functions of x, y, t and α and β are integers to be determined later, Substituting (82) into (81) and balancing the nonlinear terms against the dominant linear terms give rise to α = β = 1, u 0 = φ x, v 0 = φ y, (83) In the following, considering the Laurent series expansion of the solutions in the neighborhood of the singular manifold u = u j φ j 1, v = v j φ j 1, (84) j=0 j=0

16 Integrability 317 Substituting (84) into (81) and comparing the coefficients of (φ j 4, φ j 2 ), we have ( (j 1) 2 (j 3)φ 2 xφ y 2(j 1))(j 3)φ 3 ) ( ) x uj = 0, (85) (j 1)φ y (j 1)φ x v j from which we arrive at the resonances j = 1, 1, 1, 3.. The resonance at j = 1 represents the arbitrariness of the singularity manifold φ(x, y, t) = 0. For the other resonances, we can prove that the general solution (u(x, y, t), v(x, y, t)) of (81) admits the required number of arbitrary functions, without the introduction of any movable critical manifold, thereby satisfying the Painleve property. Thus the equation (3) passes the Painlevé test Bäcklund transformation To generate its Backlund transformation, we truncate the Laurent series at the constant level term, that is to say, u j = v j = 0 for j 2, to give rises to u(x, y, t) = u 0 φ 1 + u 1 = x ln φ + u 1, v x = u y i.e., v = 1 x u y, (86) where u 1 satisfies (3). With the aid of symbolic computation, Maple, the substitution of (86) into (81a) leads to φ t = φ xxy + 2u 1 φ xy + u 2 1φ y + u 1x φ y + 2u 1y φ x + 2u 1 x 1 u 1y φ x + x 1 u 1y φ xx, (87) Particularly, if we set u 1 = 0 in (86), we get the Cole-Hopf transformation u(x, y, t) = u 0 φ 1 + u 1 = x ln φ, (88) under which, (87) can be linearized as the third-order linear partial differential equation φ t φ xxy = 0. (89) Proposition 2. The (2+1)-dimensional GSTO-II (3) is both Painleve integrable and C-integrable. Moreover it also possesses a Backlund transformation Multi soliton-like solutions Case When u 1 = const, we propose that (87) has the x-linear formal solution N φ(x, y, t) = P (y, t) + µ i exp[θ i (y, t)x + Ψ i (y, t)], (90) i=1 where µ i = ±1, and P (y, t), Θ i (y, t) s, Ψ i (y, t) s are differentiable functions of y and t only to be determined. With the aid of symbolic computation (Maple), we substitute (90) into (87) and set to zero the coefficients of these terms x j e k(θ ix+ψ i ) to derive P t = u 2 1 P y, Θ it = Θ 2 i Θ iy + 2u 1 Θ i Θ iy + u 2 1 Θ iy, (91) Ψ it = 2Θ i Θ iy + 2u 1 Θ iy + Ψ iy (Θ i + u 1 ) 2.

17 318 Zhenya Yan Therefore we get the multi-soliton-like solutions of (3) u = Ni=1 µ i Θ i (y, t) exp[θ i (y, t)x + Ψ i (y, t)] P (y, t) + N i=1 µ i exp[θ i (y, t)x + Ψ i (y, t)] + u 1, (92) where P (y, t), Θ i (y, t), Ψ i (y, t) are defined by (91). In the following we give three special subcases to illustrate the solutions of (3) Family 1. (Multi soliton-like solutions) We give the solutions of (91) as P (y, t) = p(y + u 2 1t), Θ i (y, t) = ± y y i t t i u 1, Ψ i (y, t) = ln (y y i) αi (t t i ) 1+α i, (93) where p(y + u 2 1 t) is an arbitrary smooth function of (y + u2 1 t), and y i, t i, α i are all constants. Therefore we get the special multi-soliton-like solution of (3) u = Ni=1 µ i (± y y i t t i u 1 ) exp[(± p(y + u 2 1 t) + N i=1 µ i exp[(± y y i t t i y y i t t i u 1 )x + ln (y y i) α i (t t i ) 1+α i ] u 1 )x + ln (y y i) α i (t t i ) 1+α i ] + u 1. (94) In particular, when N = µ 1 = 1, we get the shock-like wave solution of (3) u = 1 ( ± y y ) 1 u 1 tanh 1 [( ± y y ) 1 u 1 x + ln (y y 1) α 1 (t t 1 ) 1 ] 2 t t 1 2 t t 1 (t t 1 ) α 1 p(y + u 2 1 t) and singular soliton-like solution ( ± y y ) 1 u 1 t t 1 + u 1, p(y + u 2 1t) > 0, (95a) u = 1 ( ± y y ) 1 u 1 coth 1 [( ± y y ) 1 u 1 x + ln (y y 1) α 1 (t 1 t) 1 ] 2 t t 1 2 t t 1 (t t 1 ) α 1 p(y + u 2 1 t) ( ± y y ) 1 u 1 t t 1 + u 1, p(y + u 2 1t) < 0. (95b) Family 2 (Multi soliton-like solutions) We give another set of solutions for (91) as P (y, t) = p(y + u 2 1t), Θ i (y, t) = θ i = const., Ψ i (y, t) = ψ i (y + (θ i + u 1 ) 2 t), (96) where p(y + u 2 1 t) and ψ i(y + (θ i + u 1 ) 2 t) are arbitrary smooth functions of y + u 2 1 t and y + (θ i + u 1 ) 2 t, respectively.

18 Integrability 319 Therefore we get another types of multi-soliton-like solution of (3) u = Ni=1 µ i θ i exp[θ i x + ψ i (y + (θ i + u 1 ) 2 t)] p(y + u 2 1 t) + N i=1 µ i exp[θ i x + ψ i (y + (θ i + u 1 ) 2 t)] + u 1, (97) In particular, when N = µ 1 = 1, p(y + u 2 1t) > 0, we get the shock-like wave solution of (3) u = 1 2 θ 1 tanh 1 2 [θ ix + ψ i (y + (θ i + u 1 ) 2 t ln p(y + u 2 1t)] θ 1 + u 1. (98) Family 3. (Multi-soliton solutions) From (90) and (91) we can obtain the function φ(x, y, t) has the x, y, t-linear formal solution N φ(x, y, t) = p + µ i exp[k i x + l i y + l i (k i + u 1 ) 2 t]. (99) i=1 where p, k i, l i are constants. Thus we have the multi-soliton solutions of (3) u = Ni=1 µ i k i exp(k i x + l i y + l i (k i + u 1 ) 2 t) p + N i=1 µ i exp(k i x + l i y + l i (k i + u 1 ) 2 t) + u 1. (100) Particularly,when N = p = µ 1 = 1, (100) reduces to the shock wave solution solution u = k 1 2 [tanh 1 2 (k 1x + l 1 y + l 1 (k 1 + u 1 ) 2 t) + 1] + u 1. (101) It is clearly sen that the field u is not exponentially localized in all directions. However for the two potentials q 1 = u x and q 2 = u y, that is q 1 = u x = 1 4 k2 1sech [k 1x + l 1 y + l 1 (k 1 + u 1 ) 2 t], (102a) q 2 = u y = 1 4 k 1l 1 sech [k 1x + l 1 y + +l 1 (k 1 + u 1 ) 2 t], (102b) Remark 9. It is clear to see that the solution q 1 is finite on the camber k 1 x + l 1 y + l 1 (k 1 + u 1 ) 2 t = 0 and decays exponentially away for the camber. Therefore we call this type of solution the camber soliton or camber solitary wave. Remark 10. It is easy to know that when k 1 0 and l 1 0, q 2 0. Therefore the solution q 2 is the dromion solution. Case When u 1 = u 1 (x), we suppose that (87) has the y-linear formal solution M φ(x, y, t) = Ω(x, t) + µ j exp[γ j (x, t)y + Σ j (x, t)], (103) where µ j = ±1 and Ω(x, t), Γ j (x, t), Σ j (x, t) are differential functions of x and t only to be determined.

19 320 Zhenya Yan With the aid of symbolic computation, Maple, we substitute (103) into (87) and set to zero the coefficients of these terms y j e k(γ jy+σ j ) to derive Ω t = 0 Ω(x, t) = ω(x), Γ jt = Γ jx = 0 Γ j (x, t) = γ j = const, (104) Σ jt = Γ j Σ jxx + Γ j Σ 2 jx + 2u 1Γ j Σ jx + u 2 1 Γ j + u 1x Γ j. Therefore we get the multi-soliton-like solutions of (3) u = ω (x) + M µ j Σ jx (x, t) exp[γ j y + Σ j (x, t)] ω(x) + M µ j exp[γ j y + Σ j (x, t)] + u 1 (x), (105) where ω(x) is an arbitrary smooth function of x only, and Σ j (x, t) is defined by (104). In particular, when u 1 (x) = αx + β (α, β, const), we get Σ j (x, t) = 1 2 αx2 + (k j β)x + (γ j k 2 j γ j α)t. (106) where k j, γ j are constants. Therefore we get another family of multi soliton-like solutions of (3) u = ω (x) + M µ j ( αx + k j β) exp[γ j y 1 2 αx2 + (k j β)x + γ j (k 2 j α)t] ω(x) + M µ j exp[γ j y 1 2 αx2 + (k j β)x + γ j (k 2 j α)t] +αx + β, (107) When M = µ 1 = 1 and ω(x) > 0, we have the shock-like wave solution of (3) u = 1 2 [ ] ω (x) ω(x) αx + k 1 β tanh 1 2 [γ 1y 1 2 αx2 + (k 1 β)x + γ j (k1 2 α)t ln ω(x)] Infinity many rational solutions We assume that (89) has the solution [ ω ] (x) ω(x) αx + k 1 β + αx + β, (108) m φ(x, y, t) = Ξ i (x + ky)(t t 0 ) i = Ξ 0 (x + ky) + Ξ 1 (x + ky)(t t 0 ) i=1 + + Ξ m (x + ky)(t t 0 ) m, (109) where k is a constant and Ξ is are functions of x + ky to be determined. Substituting (108) into (89) and comparing the coefficients of various power of (t t 0 ), we have ( ) 3(m+1 i) Ξ i (x + ky) = k i m m e j (x + ky) 3(m+1 i) j (m i)! m i (3(m + 1 i) j)! (t t 0) i (110)

20 Integrability 321 where e js are constants. Therefore we have the infinitely many rational solutions of (3) from (89) and (110) [ ( ) mi=1 m k i m 3(m i)+2 (m i)! m i u = [ ( ) mi=1 m 3(m+1 i) k i m (m i)! m i In particular, when m = 1, we give its rational solution ] e j (x+ky) 3(m i) j+2 (3(m i) j+2)! e j (x+ky) 3(m+1 i) j (3(m+1 i) j)! (t t 0 ) i ]. (111) (t t 0 ) i u = {[2e 1 (x + ky) + e 2 ](t t 0 ) + k 1 [ 1 12 e 1(x + ky) e 2(x + ky) e 3(x + ky) 2 +e 4 (x + ky) + e 5 ]}{[e 1 (x + ky) 2 + e 2 (x + ky) + e 3 ](t t 0 ) + k 1 [ 1 60 e 1(x + ky) e 2(x + ky) e 3(x + ky) e 4(x + ky) 2 + e 5 (x + ky) + e 6 ]} 1. (112) where t 0, e is(i = 1, 2,..., 6) are constants The mixed solutions It is clear to see that (89) has the formal solutions φ(x, y, t) = m λ i m (m i)! i=0 where C i, D i, k i, l i, λ are constants. Thus we give the solutions of (3) u = [ ( mi=0 m λ i m (m i)! mi=0 [ λ i m (m i)! where ξ i = k i x + l i y + k 2 i l it. ( m ) 3(m+1 i) m i C j (x + λy) 3(m+1 i) j t i (3(m + 1 i) j)! n + D i exp(k i x + l i y + ki 2 l i t). (113) i=1 m i ( m m i ) 3(m i)+2 ) 3(m+1 i) 6. Bi-Hamiltonian structure of equation (3) ] C j (x+λy) 3(m i) j+2 (3(m i) j+2)! C j (x+λy) 3(m+1 i) j (3(m+1 i) j)! t i + n i=1 D i k i e ξ i ] t i + n i=1 D i e ξ i. (114) Similar to the STO equation (1)[9], we assume that periodic boundary conditions for φ and u with u(x, y, t)dx = 0. In the following we will show that (3) admits a bi-hamiltonian structure[31]. It is easy to see that (89) can be written as φ t = y δh δφ, H(φ) = 1 2 φ 2 xdx (115)

21 322 Zhenya Yan From (89) we have Therefore (115) becomes φ(x, y, t) = exp( 1 x u) (116) H(u) = 1 2 u 2 (exp(2 1 x u))dx (117) We can derive from (116) and (117) δh δφ = exp( 1 x u) x δh δu (118) Differentiating (115) with respect to t once yields Substituting (115) and (118) into (119) yields u t = ( x exp( 1 x u t = x exp( 1 x u)φ t. (119) u) y exp( 1 x u) ) δh x δu = J δh δu. (120) It can be identified that J = x exp( 1 x u) y exp( 1 x u) x is a Hamiltonian operator (symplectic operator). On the other hand, it is easy to see that (89) can also be written as φ t = 3 δg x y 2 δφ, G(φ) = 1 2 φ 2 dx (121) (116) and (121) lead to δg δφ = exp( 1 x u) δg x δu Substituting (115) and (122) into (119) yields (122) where G = 1 2 u t = x exp( 1 x u) x y x exp( 1 x u) δg x δu = K δg δu, (123) exp((2 1 x u)dx. It can be proved that K = x exp( 1 x u) x y x exp( 1 x u) x. (124) is also a Hamiltonian operator. Therefore (3) possesses bi-hamiltonian structure 7. Conclusions and discussions u t = J δh δu = K δg δu. (125) In summary, we have derived the two whole hierarchies of (2) and (3) by using two Cole- Hopf transformations (4). The singularity structure analysis of (2) and (3) are carried out

22 Integrability 323 and it is shown that the two equations both have the Painleve property. The P-analysis is used to gain Backlund transformations of (2) and (3), which are used to find many typs of soliton-like solutions, soliton solutions rational solutions and other solutions. Moveover it is show that the variables, u x for the GSTO-I and u y for the GSTO-II, admit exponentially localized solutions rather than the physical field u(x, y, t) itself. Finally we give the bi- Hamiltonian structure of (3). There exist other some problems: (I) We conjecture that the whole two hierarchy of Burgers equations obtained in the section 2 also possess Painleve properties and multi-soliton solutions. (II) It may be easy to see that the other equations of the second hierarchy of odd Burgers hierarchy (16 ) also possess bi-hamiltonian structures. (III) it needs to be consider whether (2) and (3) possess other types of solutions. (IV) The symmetry reductions of (2) and (3) will be given in another papers. (V) Some properties of two (2+1)-dimensional generalized STO-Burgers equations (11c )and (19b) are also considering such as Backlund transformations, C-integrability, exact solutions, etc., (VI) We may derive (2+1)-dimensional extensions of other nonlinear evolution equations in (1+1)-dimensions by means of our generalized technique used in this paper. These will be studied further in future. References [1] M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge University Press, Cambridge, 1991). G. W. Bluman and S. Kumei, Symmetry Methods of Differential Equations (Springer, Berlin, 1989). P.J. Olver, Applications of Lie Groups to Different Equations (Spring-Verlag, New York, 1993, 2nd ed.) [2] V. E. Zakharov (ed.), What is Integrability? (Springer-Verlag, Berin, 1991). [3] M.J. Ablowitz, et al., J. Math. Phys., 21, 1006(1980). [4] J. Weiss, et al., J. Math. Phys., 24, 532 (1983). [5] H. Tasso, Report IPP (MPI fur Plasmaphysik (Garching, 6/142,1976). A.S. Sharma, H. Tasso, Report IPP ( MPI fur Plasmaphysik (Garching, 6/158, 1977) [6] P. J. Olver, J. Math. Phys., 18(1977) [7] W. Hereman, P.P. Banerjee, A. Korpel, et al., J. Phys. A, 19 (1986) 607. [8] Z. J. Yang, J. Phys. A, 27 (1994) [9] H. Tasso, J. Phys. A, 29 (1996) [10] M. Boiti, et al., Phys. Lett. A, 132(1988) 432. [11] J. Hietarinta, Phys. Lett. A, 149 (1990) 113. [12] C. Athorne and J. J. C. Nimmo, Inv. Probl., 8(1992),32. [13] R. Radla, M. Lakshmanan, J. Math. Phys., 35(1994), [14] R. Radla, M. Lakshmanan, J. Math. Phys., 38(1997), 292. [15] S. Y. Lou, J. Phys. A, 29 (1996) [16] M. Boiti, L. Martina and F. Pempinelli, Chaos, Solitons and Fractals, (1995)

23 324 Zhenya Yan [17] B. Tian and Y. T. Gao, J. Phys. A, 29 (1996) [18] Z. Y. Yan, Acta Phys. Sin., 99(1999) (in Chinese) [19] Z. Y. Yan, Commun. Theor. Phys., 36(2001)135. [20] Z. Y. Yan, Chinese. J. Phys., 40(2002) 203. [21] Z. Y. Yan, J. Phys. A, 35(2002) [22] Z. Y. Yan, Commun. Theor. Phys., 34(2000)365. [23] Z. Y. Yan and H.Q. Zhang, J. Phys. A, 34(2001)1785. [24] Z. Y. Yan and H.Q. Zhang, Comput. Math. Appl., 44(2002)1439. [25] Z. Y. Yan, Cezch. J. Phys., 53(2003) 89; 297. [26] Z. Y. Yan, Physica A (to appear). [27] Z. Y. Yan, Phys. Lett. A, (to submitted) [28] M.J. Ablowitz, et al., Stud. Appl. Math., 53(1974)249. [29] Z. Y. Yan, (to submitted) [30] F. Magri, J. Math. Phys., 19(1978) 1156.