Symbolic Computation and New Soliton-Like Solutions of the 1+2D Calogero-Bogoyavlenskii-Schif Equation

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1 MM Research Preprints, MMRC, AMSS, Academia Sinica, Beijing No., December Symbolic Computation and New Soliton-Like Solutions of the 1+D Calogero-Bogoyavlenskii-Schif Equation Zhenya Yan Key Lab of Mathematics Mechanization Institute of Systems Science, AMSS, Academia Sinica Beijing , China Abstract. In this paper many soliton-like solutions with the arbitrary functions of y and t are obtained for the 1+D Calogero-Bogoyavlenskii-Schif equation by means of some generalized ansatz and symbolic computation. Some new exact solutions are also presented from the obtained soliton-like solutions. Shock wave solution is only a special case. Keywords: Nonlinear evolution equation, symbolic computation, exact solution, solitary wave solution 1. Introduction Finding exact solutions, in particular, soliton solutions, of nonlinear evolution equations(nlees) arising from many science fields is of an important significance. There now exist many powerful methods to seek exact solutions of NLEEs. For example, Backlund transformation[1,], Darboux transformation[3,4], Hirota method[1], the homogeneous balance method[5], Lie s group method[6] and the sine-cosine method[7-11], etc. Recently, with the rapid development of computerized symbolic computation, the symbolic-computationbased method[1-14,16] and the generalized tanh method[15] become two powerful tools to find more soliton-like solutions including solitary wave solutions of many nonlinear evolution equations, for example, the KdV-type equation, the two-dimensional break soliton equation, the (+1)-dimensional dispersive long wave equation, two-dimensional Broer-Kaup equation, the (3+1)-dimensional Jimbo-Miwa equation(see Refs.[1-16] and references therein). The main idea of the symbolic-computation-based method[1-14,16] is as follows: For certain a given nonlinear partial differential equation, say, three variables x, y and t, One assumes that (1) has the solution in the form F (u t, u x, u y, u xt, u yt, u xx, u yy, u xy...) = 0 (1) u(x, y, t) = A m x n y w[z(x, y, t)] + B. where the integers m, n are to be determined, and A, B are constants. Substituting (a) into (1) and obtaining a equation of z. And then one assumes that z has the solution z(x, y, t) = P (y, t) + exp[θ(y, t)x + Ψ(y, t)]. (a) (b)

2 86 Zhenya Yan such that soliton-like solution of (1) can be found. Recently Gao and Tian[14] applied the symbolic-computation-based method to the type break soliton equation such that only a family of soliton-like solutions was obtained by using (a) and (b) with B = 0 and P (y, t) = 1. In the paper we would like to extend the transformation (a) as follows u(x, y, t) = m x n y w[z(x, y, t)] + g(x, y, t). (c) and apply it to two-dimensional Calogero-Bogoyavlenskii-Schif (CBS) Equation[17] 4u xt + 8u x u xy + 4u y u xx + u xxxy = 0. (3) with z(x, y, t) satisfying (b) such that more new soliton-like solutions that include solitary wave solutions are obtained. If we make the transformations u 1 u, t 1 4t, then (1) reduces to the breaking soliton equation[14,16,18]. The rest of this paper is organized as follows: In Sec., we obtain a Backlund transformation and Cole-Hopf transformation of (3) by using a generalized ansatz and symbolic computation. In Sec. 3, based upon the obtained Backlund transformation and Cole-Hopf transformation, many families of soliton-like solutions of (3) are found by using an ansatz and symbolic computation. Solitary wave solutions are only a special case of soliton-like solutions. Finally, some conclusions are given in Sec. 4.. Leading to a Backlund transformation According to our improved idea[16], in order to seek soliton-like solution of (3) we assume that (3) possesses the solution in the form u(x, y, t) = m x n y f[φ(x, y, t)] + g(x, y, t) = m x n y f(φ(x, y, t)) + kx + h(y, t), (4) which is a more general form of the above transformation (a), where m, n are integers to be determined later, k is a constant and h(y, t) is a function of y, t. Using the rule of leading-order analysis[5,13,16,19] and balancing the highest order linear term (i.e., u xxxy ) and nonlinear terms (i.e., u x u xy or u y u xx ), we obtain m = 1, n = 0. Thus (4) reduces to be u(x, y, t) = x f[φ(x, y, t)] + kx + h(y, t) = f (φ)φ x + kx + h(y, t). (5) where f (φ) = df/dφ. Substituting (5) into (3) and with the aid of symbolic computation, we get 4u xt + u xxxy + 8u x u xy + 4u xx u y = 4(f φ t φ x + f φ x φ xt + f φ xx φ t + f φ xxt ) +8[f f φ 4 xφ y + f φ 3 xφ xy + f φ xφ y φ xx + f f φ xφ y φ xx +f f φ x φ xy φ xx + f f φ y φ xx + f f φ xφ xxy + f φ xx φ xxy +k(f φ y φ x + f φ x φ xy + f φ xx φ y + f φ xxy )] + 4[f f φ 4 xφ y + f f φ 3 xφ xy

3 Symbolic Computation and New Soliton-Like Solutions 87 +3f φ xφ y φ xx + 3f f φ x φ xy φ xx + f f φ x φ y φ xxx + f φ xy φ xxx +h y (y, t)(f φ 3 x + 3f φ x φ xx + f φ xxx )] + f (5) φ 4 xφ y + 4f (4) φ 3 xφ xy +6f (4) φ xφ y φ xx + 1f φ x φ xy φ xx + 3f φ y φ xx + 6f φ xφ xxy + 6f φ xx φ xxy +4f φ x φ y φ xxx + 4f φ xy φ xxx + 4f φ x φ xxxy + f φ y φ xxxx + f φ xxxxy = 0. (6) To find the function f(φ), we set the coefficient of φ 4 xφ y to zero, that is, f (5) (φ) + 1f (φ)f (φ) = 0 (7) which has a solution in the form f[φ(x, y, t)] = ln φ(x, y, t), (8) Substituting(8) into (6), yields an nonlinear partial differential equation w.r.t. φ(x, y, t) (4φ xφ t φ y φ xx φ x φ xx φ xy + φ xφ xxy + φ x φ y φ xxx + 4h y (y, t)φ 3 x + 8kφ y φ x) φ(8φ x φ xt + 4φ t φ xx φ xx φ xxy + 4φ x φ xxxy + φ y φ xxxx + 1h y φ x φ xx + 16kφ x φ xy +8kφ xx φ y ) + φ (4φ xxt + φ xxxxy + 4h y φ xxx + 8kφ xxy ) = 0. Substituting (8) into (5), yields (9a) u(x, y, t) = x f[φ(x, y, t)] + kx + h(y, t) = φ x φ + kx + h(y, t). (9b) It is clear that (9a) and (9b) should be called a Backlund transformation[1] of (3). Particularly if we set k = h(y, t) = 0, then (9a) and (9b) become the Cole-Hopf transformation[] of (3). 3. Exact soliton-like solutions In what follows we mainly consider (9a). If we can obtain φ from (9a), then explicit exact solutions of (3) can be found by using (9b). When k = h(y, t) = 0, one can see that the simple trial solution of (9a), φ(x, y, t) = 1 + exp(αx + βy + γt + δ), (10) would lead to nothing but solitary waves, where α, β, γ and δ are constants. Nevertheless, this clue inspires one to proceed further more sophisticated than solitary waves. we assume that φ in (9a) is expressed by the following x-linear form φ(x, y, t) = P (y, t) + exp[θ(y, t)x + Ψ(y, t)]. (11) where P (y, t) 0, Θ(y, t) and Ψ(y, t) are differentiable functions w.r.t. y and t only to be determined later. With the aid of symbolic computation, after the substitution of (11) into (9a), we know that (9a) reduces to xe (Θx+Ψ) P Θ (4Θ t + Θ Θ y + 8kΘ y ) + e (Θx+Ψ) [Θ (4P t + Θ P y + 8kP y )

4 88 Zhenya Yan +P (8ΘΘ t 4Θ Ψ t Θ 4 Ψ y 4h y Θ kΘΘ y 8kΘ Ψ y )] +xe (Θx+Ψ) P Θ (4Θ t + Θ Θ y + 8kΘ y ) + e (Θx+Ψ) [ P Θ (4P t + Θ P y + 8kP y ) +P (8ΘΘ t + 4Θ Ψ t + 4Θ 3 Θ y + Θ 4 Ψ y + 4h y Θ kΘΘ y + 8kΘ Ψ y )] = 0. (1) Setting the coefficients of xe (Θx+Ψ), xe (Θx+Ψ), e (Θx+Ψ) and e (Θx+Ψ) to zero respectively yields a set of nonlinear partial differential equations Θ 0, P (y, t) 0, 4Θ t + Θ Θ y + 8kΘ y = 0, Θ (4P t + Θ P y + 8kP y ) + P (8ΘΘ t 4Θ Ψ t Θ 4 Ψ y 4h y Θ kΘΘ y 8kΘ Ψ y ) = 0, Θ (4P t + Θ P y + 8kP y ) + P (8ΘΘ t + 4Θ Ψ t + 4Θ 3 Θ y + Θ 4 Ψ y +4h y Θ kΘΘ y + 8kΘ Ψ y ) = 0, (13a) which further reduces to Θ 0, P (y, t) 0, 4Θ t + Θ Θ y + 8kΘ y = 0, 4P t + P y Θ + 8kP y P (4Ψ t + Θ Ψ y + ΘΘ y + 4h y Θ + 8kΨ y ) = 0. (13b) Hence from (9b), (11) and (13b) we can derive two types of soliton-like solutions for (3) Case I: When P (y, t) > 0, u 1 (x, y, t) = x f(φ(x, y, t)) + kx + h(y, t) = f φ x + kx + h(y, t) = Θ(y, t) exp[θ(y, t)x + Ψ(y, t)] + kx + h(y, t) P (y, t) + exp[θ(y, t)x + Ψ(y, t)] = 1 Θ(y, t)x + Ψ(y, t) ln P (y, t) Θ(y, t)[1 + tanh ] + kx + h(y, t). (14a) Case II: When P (y, t) < 0, u (x, y, t) = x f(φ(x, y, t)) + kx + h(y, t) = f φ x + kx + h(y, t) = Θ(y, t) exp[θ(y, t)x + Ψ(y, t)] + kx + h(y, t) P (y, t) + exp[θ(y, t)x + Ψ(y, t)] = 1 t)x + Ψ(y, t) ln P (y, t) Θ(y, t)[1 + cothθ(y, ] + kx + h(y, t). (14b) where Θ(y, t), Ψ(y, t), P (y, t), h(y, t) and k are constrained by system (13b). Remark 1: We call the above solutions solitary wave solutions if they contain the variable 1 (Θ(y, t)x + Ψ(y, t) ln P (y, t) ) which is only a linear form of x, y and t, and k = 0, h(y, t) = const.. Otherwise, we call them soliton-like solutions.

5 Symbolic Computation and New Soliton-Like Solutions 89 Remark : In what follows we focus our attention on P (y, t) > 0, however the P (y, t) < 0 case is analogous to the P (y, t) > 0 case. The physical interest of the above mentioned solutions lies in the fact that they describe certain soliton-like surface waves. The actual form of the amplitude depends on the choices of Θ(y, t), Ψ(y, t) and P (y, t), while its horizontal velocity depends on Θ(y, t). As examples, we now consider further the following several cases: Case 1: When Θ = θ = const 0, it is easy to obtain from system (13b) that 4P t + (8k + θ )P y P [4Ψ t + (8k + θ )Ψ y + 4θh y ] = 0. (15) and thus soliton-like solutions of (3) are as follows u(x, y, t) = 1 θx + Ψ(y, t) ln P (y, t) θ[1 + tanh ] + kx + h(y, t), P (y, t) > 0. (16) where P (y, t), k, h(y, t) and Ψ(y, t) satisfy (15). From (15), we may obtain many solutions for these functions P, Ψ and h. In the following we will give several cases: Case 1a: (Soliton-like solution) If we further assume that P (y, t) = p(4y (8k + θ )t), then we have from (15) that Ψ(y, t) = ψ(z), h(y, t) = c 1(t) t 4θ ψ(z) + c (t), z = 4y (8k + θ )t + c1 (s)ds, (17) where p( ) is an arbitrary function of 4y (8k + θ )t, ψ(z) is an arbitrary function of 4y (8k + θ )t + t c 1 (s)ds, and c 1 (t), c (t) are two arbitrary functions of t. Thus when p[4y (8k + θ )t] > 0, we have the soliton-like solutions of (3) u(x, y, t) = 1 θ[1 + tanh θx + ψ[4y (8k + θ )t + t c 1 (s)ds] ln[p(4y (8k + θ )t)] ] +kx c 1(t) t 4θ ψ[4y (8k + θ )t + c1 (s)ds] + c (t) (18) In particular, when P (y, t) = p 0 = const > 0, k = 0, h(y, t) = h 0, we have from (15) that Ψ(y, t) = c 3 (4y θ t) + c 4. Thus we have the shock wave solution of (3) u(x, y, t) = 1 θ[1 + tanh θx + c 3(4y θ t) + c 4 ln p 0 ] + h 0. (19) where c 3, p 0, h 0, c 4 are constants. Case 1b: (Soliton-like solution)

6 90 Zhenya Yan If we further assume that P (y, t) = exp(p 1 y + p t), then we have from (15) that Ψ(y, t) = ψ(z), h(y, t) = 4p + (8k + θ )p 1 y + h(t), z = 4y (8k + θ )t, (0) 4θ where ψ(z) is an arbitrary function of 4y (8k + θ )t, h(t) an arbitrary function of t, and p 1, p constants. Thus we have the soliton-like solution of (3) u(x, y, t) = 1 θ[1 + tanh θx + ψ[4y (8k + θ )t] (p 1 y + p t) ] +kx + 4p + (8k + θ )p 1 y + h(t). (1) 4θ Remark 3: We only give two special solutions of (15) such that exact solutions of (3) are obtained. We think (15) also possesses other types of solutions which will lead to other types of solutions of (3). Case : When k = 0, we have from (13b) that { 4Θt + Θ Θ y = 0, 4P t + P y Θ = P (4Ψ t + Θ Ψ y + ΘΘ y + 4h y Θ). () Case a:(soliton-like solution) From () we get the solutions Θ(y, t) = ±, P (y, t) = p( ), h(y, t) = c 5 + h(t), Ψ(y, t) = β ln( ) (1 ± c 5 + β) ln(y + y 0 ), where p( ) is an arbitrary function of y+y0 and β, c 5 constants. Thus we have the solution of (3) u(x, y, t) = ± tanh ± y+y0 (t+t x + ln 0 ) β ln[p( y+y0 (y+y 0 ) 1±c 5 +β )] +(c 5 ± 1) + h(t). (3) Case b: (Soliton-like solution) From () we get another solutions Θ(y, t) = ±, P (y, t) = p( ), h(y, t) = + h(t), Ψ(y, t) = ψ( ). (4)

7 Symbolic Computation and New Soliton-Like Solutions 91 where p( ) and ψ( ) are both arbitrary functions of y+y0, h(t) an arbitrary function of t. Thus we have the soliton-like solution of (3) u(x, y, t) = ± tanh ± Case c:(soliton-like solution) From () we get another solutions y+y0 x + ψ( y+y0 ) ln[p( Θ(y, t) = ±, P (y, t) = p( ), h(y, t) = 1 y+y0 )] + h(t). (5) y s + y0 ψ (s)ds + h(t), Ψ(y, t) = ψ(y), (6) where p( ) is an arbitrary functions of y+y0, ψ( ) an arbitrary function of y and h(t) an arbitrary function of t. Thus we have the soliton-like solution of (3) u(x, y, t) = ± tanh ± 1 Case d: (Soliton-like solution) From () we get another solutions y+y0 x + ψ(y) ln[p( y+y0 )] y s + y0 ψ (s)ds + h(t). (7) Θ(y, t) = ±, P (y, t) = p( ), h(y, t) = (y + y 0 )( )ψ (t) + h(t), Ψ(y, t) = ψ(t), (8) where p( ) is an arbitrary functions of y+y0, ψ( ) an arbitrary function of t and h(t) an arbitrary function of t. Thus we have the soliton-like solution of (3) u(x, y, t) = ± tanh ± y+y0 x + ψ(t) ln[p( y+y0 )] (y + y 0 )( )ψ (t) + h(t). (9) Remark 4: Here we only give four special solutions of () such that some exact solutions of (3) are obtained. We think () also possesses other types of solutions which will

8 9 Zhenya Yan lead to other types of solutions of (3). 4. Summary and Question In summary, we have found several families of exact soliton-like solutions including new exact solutions of the 1+D Calogero-Booywlenskii-Schif equation by using a generalized transformation and symbolic computation. The method is powerful to find more soliton-like solution of NLEEs and may also be extended to other nonlinear partial differential equations arising from many fields such as physics, mechanics, optics, etc. For example, if we applied the transformation (4) to the typical break soliton equation[14], then we would obtain more soliton-like solutions than ones by Gao and Tian[14]. In addition, a natural problem is whether (3) has other types of exact solutions, in particular, soliton solutions, which need to be considered further by seeking new approaches. Acknowledgements The author is grateful to the anonymous referee for some valuable suggestions. project was supported by the PIMS Postdoctoral Fellowship. The References [1] M.R. Miura, Backlund Transformation, (Springer-Verlag, Berlin, 1978 ). [] M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge University Press, Cambridge, 1991). [3] C.H. Gu et al., Soliton Theory and its Applications (Springer-Verlag, Berlin, 1995). [4] V. B. Matveev and M. A. Salle, Darboux transformations and Solitons (Springer, Berlin, 1991) [5] M.L. Wang, Phys. Lett. A, 199,169(1995). [6] P. J. Olver, Applications of Lie Group to Differential Equations, Springer-Verlag, Berlin (1986). G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer-Verlag, Berlin (1989). [7] C. T. Yan, Phys. Lett. A, 4(1996) 77. [8] Z. Y. Yan and H. Q. Zhang, Phys. Lett. A, 5(1999) 91. [9] Z. Y. Yan et al., Acta Phys. Sin., 48(1999) 1. [10] Z. Y. Yan and H. Q. Zhang, Appl. Math. Mech., 0, 385 (000). [11] Z. Y. Yan and H. Q. Zhang, Phys. Lett. A, 85(001) 355. Z. Y. Yan, Phys. Lett. A, 9(001)100. [1] Y. Gao and B. Tian, Comput. Math. Applic. 30 (1995) 97. [13] B. Tian and Y. Gao, J. Phys. A 9(1996) 895. [14] Y. Gao and B. Tian, Comput. Math. Applic. 33 (1997) 115. [15] W. Hong and K. Oh, Comput. Math. Applic. 39 (000) 9. [16] Z. Y. Yan and H. Q. Zhang, J. Phys. A, 34(001) Z. Y. Yan, J. Phys. A, 35(00) 993. Z. Y. Yan and H. Q. Zhang, Comput. Math. Appl. (to appear). [17] P. G. Estvez and G. A. Hernaez, J. Phys. A, 33(000)131.

9 Symbolic Computation and New Soliton-Like Solutions 93 [18] F. Calogero and A. Degasperis, Nuovo Cimento B, 3(1976) 01. [19] J. Weiss et al., J. Math. Phys. 4(1983) 5.