New explicit solitary wave solutions for (2 + 1)-dimensional Boussinesq equation and (3 + 1)-dimensional KP equation

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1 Physics Letters A 07 (00) New explicit solitary wave solutions for ( + 1)-dimensional Boussinesq equation and ( + 1)-dimensional KP equation Yong Chen, Zhenya Yan, Honging Zhang Department of Applied Mathematics, Dalian University of Technology, Dalian 1104, People s Republic of China Received November 001; received in revised form 1 September 00; accepted November 00 Communicated by A.R. Bishop Abstract In this Letter, we study ( + 1)-dimensional Boussinesq equation and ( + 1)-dimensional KP equation by using the new generalized transformation in Homogeneous Balance Method (HBM). As a result, many explicit exact solutions, which contain new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions and periodic wave solutions, are obtained. 00 Elsevier Science B.V. All rights reserved. Keywords: Boussinesq equation; KP equation; HBM; Riccati equation; Solitary wave solution; Periodic wave solution 1. Introduction In recent years, searching for explicit exact solutions, in particular, solitary wave solutions, of nonlinear evolution equations (NEEs) in mathematical physics plays an important role in soliton theory [1 11,14, 1]. Particularly, various powerful methods have been presented, such as, Backlund transformation, Darboux transformation, Cole Hopf transformation, tanh method, sine cosine method, Painlevé method, homogeneous balance method (HBM), Hirota method [1], Lie group analysis, similarity reduced method and so on. Based upon the well-known Riccati equation, homogeneous balance method (HBM) proposed by Wang et al. [,7] is to find exact solutions of certain nonlinear PDEs. Fan and Zhang [1,15] improved considerably the key steps of the HBM. Particularly, more general ansatz have been proposed in order to obtain new form of solutions. Recently, Senthilvelan [1] studied the travelling wave solutions for ( + 1)-dimensional Boussinesq equation and ( + 1)-dimensional KP equation by homogeneous balance method (HBM) and explored certain new solution of the equations. In this Letter, we would like to discuss further ( + 1)-dimensional Boussinesq equation and ( + 1)- dimensional KP equation by our improved method, in which we presented a new generalized transformation [17]. As a result, more new exact solutions, which include the solutions obtained by Senthilvel [1], are obtained. * Corresponding author. addresses: chenyong@dlut.edu.cn (Y. Chen), yanzy@student.dlut.edu.cn (Z. Yan) /0/$ see front matter 00 Elsevier Science B.V. All rights reserved. doi:10.101/s (0)018-7

2 108 Y. Chen et al. / Physics Letters A 07 (00) Method Our method is summed up as follows. For the given nonlinear evolution equations, say, in three variables, x,y,t F(u,u u x,u y,u xu yu tu xy,u yy,...)= 0, we seek the following formal travelling wave solutions (1) u(x, t) = u(ξ), ξ = x + βy λt, () where, β, λ are all constants to be determined later. Then () reduces to a nonlinear ordinary differential equation F 0 (u, u,u,...)= 0, () where denotes dξ d. In order to seek the travelling wave solutions of (), we take the following transformations u(ξ) = m i=1 ω i 1 ( (ξ) [a i ω(ξ) + a i µ µ ω (ξ) ) ] + a 0, and the new variable ω = ω(ξ) satisfying ω R ( 1 + µ ω ) = dω (5) dξ R( 1 + µ ω ) = 0, where µ j =±1 (j = 1, ); m is an integer to be determined and a i,a i (i = 0, 1,,...,m; j = 0, 1,,...,n),R are constants to be determined later. There exist the following steps to be considered further. Step 1. Determine the values of m of (4) by respectively balancing the highest order partial derivative term and the nonlinear term in (), it is easy to get the value of m. Step. With the aid of MATHEMATICA, substituting system (4) along with the condition (5) into (), yields a system of algebraic equations w.r.t. ω i (µ 1 + µ 1 µ ω ) j/ (j = 0, 1; i = 0, 1,,...). Step. Collect all terms with the same power in ω i (µ 1 + µ 1 µ ω ) j/ (j = 0, 1; i = 0, 1,,...). Setting the coefficients of the terms ω i (µ 1 + µ 1 µ ω ) j/ (j = 0, 1; i = 0, 1,,...) to zero their coefficients to get a over-determined system of nonlinear algebraic equations w.r.t. the unknown variables λ,,β,r,a 0,a j,b j (i = 1,,...,m). Step 4. With the aid of MATHEMATICA, we apply Wu-elimination method [18,19] to solve the above overdetermined system of nonlinear algebraic equations obtained in Step 4, yields the values of λ,,β,r,a 0,a j,b j (i = 1,,...,m). Step 5. It is well known that the general solutions of (5) are (4) (1) When taking µ = 1, 1 forb = 0, ω = ω(ξ) = A B exp( Rξ) A + B exp( Rξ) = 1 fora = 0, tanh [ Rξ 1 ln( )] A B for AB > 0, coth [ Rξ 1 ln( B A )] for AB < 0. ()

3 Y. Chen et al. / Physics Letters A 07 (00) When A,B are arbitrary constants satisfying A + B 0. This solution may be obtained by three tricks: a Möbius transformation, a Cole Hopf transformation or a relation (ω 1 ω )(ω ω 4 ) (ω 1 ω )(ω ω 4 ) = C = const of the solutions ω i, 1 i 4, beginning with three known solutions 1, 1, tanh(rξ). () When µ = 1, { tan(rξ + ξ0 ), ω = ω(ξ) = (7) cot(rξ + ξ 0 ). Thus according to (), (), (), (7) and the conclusions in Step 4, we can obtain more travelling wave solutions of ().. Solitary wave solution and periodic wave solution.1. ( + 1)-dimensional Boussinesq equation Let us consider a ( + 1)-dimensional generalization of Boussinesq equation [0] u tt u xx u yy ( u ) xx u xxxx = 0. According to the above steps, we firstly make the following formal travelling wave transformation: u(x,y,t) = u(ξ), ξ = x + βy λt, where, β, λ are constants to be determined. Substituting (9) into (8) and integrating it twice reads 4 u + u + ( + β λ ) u = 0. According to Step 1 in Section, we support that (10) has the following formal solutions ( u = a 0 + a 1 ω + b 1 µ µ ω ) + a ω ( + b ω µ µ ω ) (8) (9) (10) (11) and ω = ω(ξ) satisfying Eq. (5), where a 0,a 1,a,b 1,b are constants to be determined later. With the aid of MATHEMATICA, substituting (11) into (10) along with (5) and collecting all terms with the same power in ω i (µ 1 + µ 1 µ ω ) j/ (j = 0, 1; i = 0, 1,,, 4), yield a system of equations w.r.t. ω i (µ 1 + µ 1 µ ω ) j/. Setting the coefficients of ω i (µ 1 + µ 1 µ ω ) j/ (j = 0, 1; i = 0, 1,,, 4) in the obtained system of equations to zero, we can deduce the following set of over-determined algebraic polynomials with the respect the unknowns a R 4 + ( a0 + b 1 µ 1) ( + a 0 λ + β ) = 0, a 1 µ R 4 + (a 0 a 1 + b 1 b µ 1 ) ( + a 1 + β λ ) = 0, 8a µ R 4 + ( a1 + a 0a + b µ 1 + b1 µ ) 1µ ( + a + β λ ) = 0, a 1 R 4 + (a 1 a + µ 1 µ b 1 b ) = 0, a R 4 + ( a + µ 1µ b) = 0, (1.1) (1.) (1.) (1.4) (1.5)

4 110 Y. Chen et al. / Physics Letters A 07 (00) b 1 µ R 4 + a 0 b 1 ( + b 1 + β λ ) = 0, 5b µ R 4 + (a 1 b 1 + a 0 b ) ( + b + β λ ) = 0, b 1 R 4 + (a b 1 + a 1 b ) = 0, b R 4 + a b = 0. From which we have (1.) (1.7) (1.8) (1.9) Case 1 a 0 = ( 4µ ± )R, a 1 = b 1 = b = 0, a = R, λ= [( β + ) ± 4R ] 1/. Case a 0 = ( 5µ ± 1)R, a 1 = b 1 = 0, a = R, b =± R, µ1 µ λ = [( β + ) ± R ] 1/. Therefore according to Step 5, eight families of explicit and exact travelling wave solutions, which contain solitary wave solutions, periodic wave solutions and new travelling wave solutions, are found as follows for () u 1 = (4 ± )R R tanh [ R ( x + βy [( β + ) ± 4R ] 1/ t ) ], u = (4 ± )R R coth [ R ( x + βy [( β + ) ± 4R ] 1/ t ) ], u = ( 4 ± )R R tan [ R ( x + βy [( β + ) ± 4R ] 1/ t ) ], u 4 = ( 4 ± )R R cot [ R ( x + βy [( β + ) ± 4R ] 1/ ) ] u 5 = (5 ± 1)R R { tanh [ R(x + βy λt) ] i tanh [ R(x + βy λt) ] sech [ R(x + βy λt) ]}, u = (5 ± 1)R R { coth [ R(x + βy λt) ] coth [ R(x + βy λt) ] csch [ R(x + βy λt) ]}, u 7 = ( 5 ± 1)R R { tan [ R(x + βy λt) ] tan [ R(x + βy λt) ] sec [ R(x + βy λt) ]}, u 8 = ( 5 ± 1)R R { cot [ R(x + βy λt) ] cot [ R(x + βy λt) ] csc [ R(x + βy λt) ]}, where λ =[(β + ) ± R ] 1/. Remark 1. It is easily seen that u 1,u,u,u 4 are just the solution () and () by Senthilvelan [1]. But to our knowledge, the obtained solutions of (1), u 5,u,u 7,u 8 were not found before. M. Chen obtained many exact solution of various Boussinesq systems by the method presented in [1], due to our more generalized transformation than the ansatz in [1], so by our method we can recover the solutions in [1].

5 Y. Chen et al. / Physics Letters A 07 (00) ( + 1)-dimensional KP equation Let us now consider the ( + 1)-dimensional KP equation u xt u x + uu xx u xxxx u yy u zz = 0. According to the same as the above-mentioned steps, we firstly make the following formal travelling wave transformation: (1) u(x,y,t) = u(ξ), ξ = x + βy + γz λt, where, β, γ, λ are constants to be determined. Substituting (14) into (1) gives rise to 4 u u + ( + γ + λ ) u = 0. We assume that (15) has the solution in the form ( u = a 0 + a 1 ω + b 1 µ µ ω ) + a ω ( + b ω µ µ ω ) (14) (15) (1) and ω = ω(ξ) satisfying Eq. (5), where a 0,a 1,a,b 1,b are constants to be determined later. Substituting (1) into (15) along with (5), we can obtain a system of over-determined algebraic polynomials a R 4 + ( a0 + b 1 µ 1)( ) ( + a 0 λ + γ + β ) = 0, a 1 µ R 4 + (a 0 a 1 + b 1 b µ 1 ) ( ) ( + a 1 λ + γ + β ) = 0, 8a µ R 4 + ( a1 + a 0a + b µ 1 + b1 µ )( 1µ ) ( + a λ + γ + β ) = 0, a 1 R 4 + (a 1 a + b 1 b µ 1 µ ) ( ) = 0, a R 4 + ( a + b µ )( µ 1 ) = 0, b 1 µ R 4 a 0 b 1 ( + b 1 λ + γ + β ) = 0, 5b µ R 4 + (a 1 b 1 + a 0 b ) ( ) ( + b λ + γ + β ) = 0, b 1 R 4 + (a b 1 + a 1 b ) = 0, b R 4 ( + a b ) = 0, from which we can obtain (17.1) (17.) (17.) (17.4) (17.5) (17.) (17.7) (17.8) (17.9) Case 1 a 0 = 4µ R ± R, a 1 = b 1 = 0, a = R, λ= (β + γ ) ± 4R 4.

6 11 Y. Chen et al. / Physics Letters A 07 (00) Case a = R, a 1 = b 1 = 0, b =± R, a 0 = 5µ R ± R, µ1 µ λ = (β + γ ) ± R 4. Thus we can find eight families of explicit and exact travelling wave solutions, which contain solitary wave solutions, periodic wave solutions and new travelling wave solutions, are found as follows for (1) u 1 = 4R ± R + R tanh (x + βy + γz (β + γ ) ± 4R 4 )] u = 4R ± R + R coth (x + βy + γz (β + γ ) ± 4R 4 )] u = 4R ± R + R tan (x + βy + γz (β + γ ) ± 4R 4 )] u 4 = 4R ± R + R cot (x + βy + γz (β + γ ) ± 4R 4 )] u 5 = 5R ± R + R [ tanh [ R(x + βy + γz λt) ] ± i tanh [ R(x + βy + γz λt) ] u = 5R ± R u 7 = 5R ± R u 8 = 5R ± R where λ = (β +γ )±R 4. sech [ R(x + βy + γz λt) ]], + R [ coth [ R(x + βy + γz λt) ] ± coth [ R(x + βy + γz λt) ] csch [ R(x + βy + γz λt) ]], + R [ tan [ R(x + βy + γz λt) ] ± tan [ R(x + βy + γz λt) ] sec [ R(x + βy + γz λt) ]], + R [ cot [ R(x + βy + γz λt) ] ± cot [ R(x + βy + γz λt) ] csc [ R(x + βy + γz λt) ]], Remark. It is easily seen that u 1,u,u,u 4 are just the solution (8) and (9) by Senthilvelan [1]. But to our knowledge, the obtained solutions of (1), u 5,u,u 7,u 8 were not found before. 4. Conclusions In summary, based on the well-known Riccati equation, many new types of exact solutions for both ( + 1)- dimensional Boussinesq equation and ( + 1)-dimensional KP equation have been derived by a generalized

7 Y. Chen et al. / Physics Letters A 07 (00) transformation. These solutions contain the known ones [1]. Seven kinds of them are singular soliton solutions. Such solutions develop a singularity at a finite point, i.e., for any fixed t = t 0,thereexistx 0 at which these solutions blow up. There is much current interest in the formation of so-called hot spots or blow up of solutions. It appears that these singular solutions will model this physical phenomena. The method can be also easy to be extended to other NEEs and is sufficient to seek more new solitary wave solutions of NEEs. It not only uses a more generalized transformation to produce a overdetermined system of nonlinear algebraic equation but also can look for more solutions. In addition, this method is also computerizable, which allow us to perform complicated and tedious algebraic calculation on a computer. Acknowledgements The work is supported by the National Natural Science Foundation of China under the Grant No , the National Key Basic Research Development Project Program under the Grant No. G and Doctoral Foundation of China under the Grant No References [1] M.J. Ablowitz, P.A. Clarkson, Soliton, Nonlinear Evolution Equations and Inverse Scatting, Cambridge Univ. Press, New York, [] C.H. Gu, et al., Soliton Theory and its Application, Zhejiang Science and Technology Press, Zhejiang, [] D. Cox, et al., Ideal, Varieties and Algorithms, Springer-Verlag, New York, [4] M.J. Ablowitz, et al., J. Math. Phys. 0 (1989) 0. [5] C.T. Yan, Phys. Lett. A 4 (199) 77. [] M.L. Wang, Phys. Lett. A 1 (199) 7. [7] M.L. Wang, Phys. Lett. A 199 (1995) 19. [8] Z.Y. Yan, H.Q. Zhang, Phys. Lett. A 5 (1999) 91. [9] Z.Y. Yan, H.Q. Zhang, Appl. Math. Mech. 1 (000) 8. [10] W.X. Ma, et al., Int. J. Non-Linear Mech. 1 (199) 9. [11] E.G. Fan, H.Q. Zhang, Appl. Math. Mech. 19 (1998) 71. [1] M. Lakshmana, R. Raadha, Pramana J. Phys. 48 (1997) 1, and references therein. [1] E. Fan, H. Zhang, Phys. Lett. A 45 (1998) 89. [14] G.B. Whitham, Proc. R. Soc. London, Ser. A 99 (197). [15] E. Fan, H. Zhang, Phys. Lett. A 4 (1998) 40. [1] M. Senthilvelan, Appl. Math. Comput. 1 (001) 81. [17] Z.Y. Yan, H.Q. Zhang, Phys. Lett. A 85 (001) 55. [18] W. Wu, Kexue Tongbao 1 (198) 1. [19] W. Wu, in: D.Z. Du, et al. (Eds.), Algorithams and Computation, Springer, Berlin, 1994, p. 1. [0] M.A. Allen, G. Rowlands, Phys. Lett. A 5 (1997) 145. [1] M. Chen, Appl. Math. Lett. 11 (1998) 45.

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