Double Elliptic Equation Expansion Approach and Novel Solutions of (2+1)-Dimensional Break Soliton Equation

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1 Commun. Theor. Phys. (Beijing, China) 49 (008) pp c Chinese Physical Society Vol. 49, No., February 5, 008 Double Elliptic Equation Expansion Approach and Novel Solutions of (+)-Dimensional Break Soliton Equation SUN Wei-Kun,,3 CAO Nan-Bin,,3 and SHEN Ya-Liang 3,4, Department of Mathematics and Physics, Tianjin University of Technology and Education, Tianjin 300, China School of Science, Shijiazhuang University of Economics, Shijiazhuang 05003, China 3 Key Laboratory of Mathematics Mechanization, Academy of Mathematics and System Sciences, the Chinese Academy of Sciences, Beijing 00080, China 4 School of Science, Natong University, Nantong 609, China (Received March 9, 007; Revised May 4, 007) Abstract In this paper, by means of double elliptic equation expansion approach, the novel double nonlinear wave solutions of the (+)-dimensional break soliton equation are obtained. These double nonlinear wave solutions contain the double Jacobi elliptic function-like solutions, the double solitary wave-like solutions, and so on. The method is also powerful to some other nonlinear wave equations in (+) dimensions. PACS numbers: 0.30.Ik, Yv Key words: break soliton equation, symbolic computation, double elliptic equations, double soliton-like solution, nonlinear wave solution Introduction It is known that the exact solutions of nonlinear wave equations may describe not only the propagation of nonlinear waves but also spatially localized structures of permanent shape that may be of interest to experiments. [] Since the inverse scattering transformation (I.S.T.) was presented, [] there has been increasing interesting in searching for new nonlinear wave equations and the related issue of the construction of exact solutions to a wide class of nonlinear wave equations in soliton and integrable system theory. Up to now, many powerful methods have been developed such as inverse I.S.T. method, [,] Bäcklund transformation method, [,3,4] Hirotas bilinear method, [5] homogeneous balance method, [6] the tanh method, [7] the separation of variables method, [8] the rational expansion method, [9,0] the F -expansion method, [] the theta function method, [] and other direct ansatz methods. [3 7] Special exact solutions of nonlinear evolution equations may be found by using direct ansatz methods. [7 7] To construct the proper ansatz, a clue may be given from Painlevé analysis, which is based on seeking solutions whose movable critical points are poles only. Thus the use of elliptic function functions [0,,3] in the ansatz is rather natural because they are the most general functions having such singular points and has the relations with nonlinear equations. Up to now, in the ansatz, four theta functions, [] three Jacobian elliptic functions, [] four Jacobian elliptic functions, [3] had been used. Jacobian elliptic functions satisfy elliptic equation. In fact, there are twelve Jacobian elliptic functions including Jacobian elliptic sine function, Jacobian elliptic cosine function, Jacobian elliptic and function of the third as well as nine Jacobian elliptic functions defined by Glaisher. In Refs. [9] and [0], Chen and Wang proposed the rational expansion method to construct the rational exact solutions of nonlinear wave equations and obtain much success. Subsequently, Wang proposed an efficient algebraic method [4] to construct double solitary wave solutions of a class of nonlinear wave equations. In this paper, based on Wang s ideas, [,4] we present the double elliptic equation expansion method to derive the double nonlinear wave solutions of the (+)-dimensional nonlinear wave equations. We take the (+)-dimensional break soliton equation as an example to illustrate the validity of the method. The (+)-dimensional break soliton equation [7 0] reads u t + λu xxy + 4λuv x + 4λu x v = 0, u y = v x, () where λ is a nonzero constant, which describes the (+)- dimensional interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis, and it seems to have been generated. [0] Li and Zhang [8] have constructed infinitely many symmetries for Eq. () via the infinitesimal version of the dressing method. Recently, Yan [9] and Chen et al. [7] obtained the exact soliton-like solutions of Eq. () by direct ansatz methods and symbolic computation. The rest of this paper is organized as follows. In Sec., the double elliptic equation expansion method is presented. The method of Sec. is applied to the (+)- dimensional break soliton equation and families of double nonlinear wave solutions are constructed in Sec. 3. Conclusions are given in the last section. The project supported partially by the 973 Project under Grant No. 004CB shen.yl@ntu.edu.cn

2 8 SUN Wei-Kun, CAO Nan-Bin, and SHEN Ya-Liang Vol. 49 Double Elliptic Equation Expansion Method In what follows we establish the double elliptic equation expansion method. Step For a given nonlinear equation with three independent variables x, y, and t and a dependent variable u = u(x, y, t): F (u t, u x, u y, u xt, u yt, u tt, u xy,...) = 0. () where t is a time variable, x and y are spatial variables, and the subscripts t, x, and y refer to the partial derivatives with respect to t, x, and y, respectively. We first seek the solutions of Eq. () by the special ansätz n u(x, y, t) = A 0 + A j i f i (ξ)g j (η), (3) k= i+j=k where ξ, η, A 0, and A j i (i, j =,,..., n) are the functions of (x, y, t) to be determined, n is an undetermined integer, and φ = φ(ξ), ψ = ψ(η) satisfy elliptic equations, f (ξ) = l f 4 (ξ) + m f (ξ) + n, g (η) = l g 4 (η) + m g (η) + n, (4) where l 0, m, n, l 0, m and n are real parameters. Step To seek the solutions of form Eq. (3), we must first determine the leading n exponents. By balancing the highest order derivative term and the highest order nonlinear term, we can determine the value of n in Eq. (3). Step 3 With the aid of symbolic computation, substituting Eq. (3) with Eq. (4) into Eq. (), yields a partial differential equation about f i (ξ)g j (η) (i = 0,,,... ; j = 0,,,...). Step 4 Derive a set of over-determined partial differential equations. Because the terms f i (ξ)g j (η) (i = 0,,,... ; j = 0,,,...) are linearly independent, setting the coefficients of the terms f i (ξ)g j (η) (i = 0,,,... ; j = 0,,,...) to zero, a series of over-determined partial differential equations about the unknown variables ξ, η, A 0, and A j i (i, j =,,..., n) can be obtained. Step 5 Solving the nonlinear parameterized overdetermined partial differential equations, the expressions of ξ, η, A 0, and A j i (i, j =,,..., n) are determined. Step 6 Substituting the expressions of ξ, η, A 0, and A j i (i, j =,,..., n) in Step 5 into Eq. (3), the general form of non-travelling wave solutions of Eq. () are derived. Subsequently, substituting the explicit solutions of Eq. (4) into the general form of solutions, the exact non-travelling wave solutions of Eq. () are obtained. Finally, check the solutions by substituting them into initial Eq. (). Remark Chen and Wang s rational expansion method [9,0] is powerful to construct complex rational form of solutions of nonlinear wave equations, but it cannot be used to construct double nonlinear wave solutions. Here we further extend Wang et al. s method, [4] which is based on the ideas of Chen and Wang [9,0] in order to obtain more new forms of exact solutions of soliton equations. Through the detailed description above, it is easy to see that compared with the Chen and Wang et al. s method, [9,0,4] the above method is more powerful to construct the double non-travelling wave solutions of some nonlinear wave equations. In the following, we will list the explicit solutions of elliptic equation, f (ξ) = l f 4 (ξ) + m f (ξ) + n, (5) which is a very important auxiliary equation to construct exact solutions of nonlinear wave equations. It is well known that the relations among the values of l, m, and n and the corresponding solutions of Eq. (5) are given as in Table. Table The relations among the values of l, m, m, and n, and the corresponding solutions of Eq. (5). l m n f k k sn(ξ), cd(ξ) k k k cn(ξ, k) k k dn(ξ, k) k k k nc(ξ, k) k k sc(ξ, k) k k cs(ξ, k) k k ( k ) ds(ξ, k) k k dc(ξ, k), ns(ξ, k) + k k nd(ξ, k) k ( k ) k sd(ξ, k) k /4 k / k /4 sn(ξ, k) + i, k cn(ξ, k) /4 k / k /4 ns(ξ, k) + ds(ξ, k) /4 / k /4 ns(ξ, k) + cs(ξ, k) ( k )/4 ( + k )/ ( k )/4 nc(ξ, k) + sc(ξ, k) In Table k is the modulus of Jacobian elliptic functions, which is an intrinsic parameter of the solution families when it is used to express the solutions of soliton equations. In what follows, for convenience, we denote sn(ξ, k) to be sn(ξ), cn(ξ, k) to be cn(ξ), and so on. The relationships among Jacobian elliptic functions are denoted as follows: [] ns(ξ) = sn(ξ), nc(ξ) = cn(ξ), nd(ξ) = dn(ξ), cd(ξ) = cn(ξ) dn(ξ), cs(ξ) = sn(ξ) cn(ξ), dn(ξ) dc(ξ) = cn(ξ), dn(ξ) ds(ξ) = sn(ξ), cn(ξ) sc(ξ) = sn(ξ), sn(ξ) sd(ξ) = dn(ξ). (6) It is known that when k and k 0 the Jacobian elliptic functions can degenerate into hyperbolic functions and trigonometric functions. The concrete expressions are as follows. When k, sn(ξ) tanh(ξ), cn(ξ) sech(ξ),

3 No. Double Elliptic Equation Expansion Approach and Novel Solutions of (+)-Dimensional Break Soliton Equation 83 When k 0, dn(ξ) sech(ξ). (7) sn(ξ) sin(ξ), cn(z) cos(ξ), dn(z), sc(ξ) tan(ξ). (8) In addition, the Jacobian elliptic functions satisfy the following relations: sn (ξ) + cn (ξ) =, k sn (ξ) + dn (ξ) =, + k sd (ξ) = nd (ξ), (9) sc (ξ) + = nc (ξ), ns (ξ) = + cs (ξ), ns (ξ) = m + ds (ξ). (0) 3 Double Nonlinear Wave Solutions of (+)-Dimensional Break Soliton Equation In what follows, we will apply the double elliptic equation expansion method in Sec. to Eq. () to illustrate the validity of the method and construct the double nonlinear wave solutions of Eq. (). By balancing the highest order partial derivative term and the nonlinear term of Eq. (), we come to n =. So assume that equation () owns the following form of solution, u = a 0 + a f(ξ) + b g(η) + a f (ξ) + b f(ξ)g(η) + a 3 g (η), v = c 0 + c f(ξ) + d g(η) + c f (ξ) + d f(ξ)g(η) + c 3 g (η), () where ξ = α x + p + q, η = α x + p + q, a 0, a, b, a, b, c 0, d, c, c, d, a 3, and d 3 are the functions of (y, t) to be determined, α and α are undetermined constants, p and p are functions of y to be determined, q and q are functions of t to be determined and f = f(ξ), g = g(η) satisfy Eq. (4). With the aid of symbolic computation, substituting Eq. () along with Eq. (4) into Eq. () yields a set of partial differential equations about f i (ξ)g j (η) (i = 0,,,... ; j = 0,,,...). Then vanishing the coefficients of the terms f i (ξ)g j (η) (i = 0,,,... ; j = 0,,,...) yields a set of over-determined partial differential equations with respect to a 0, a, b, a, b, c 0, d, c, c, d, a 3, c 3, α, α, p, p, q, and q. Solving the over-determined partial differential equations, the following results are obtained: ( ) α c 0 = m + αm dp /dy + F (t), p = p, p = α p + C, c 3 = 3(dp /dy)αl, α α q = 4λ α F (t)dt + C 3, c = 3 ( α dp ) l, b = a = 0, a 3 = 3 dy α l, a = 3 α l, b = c = d = d = 0, a 0 = α m α m, q = 4λ F (t)α dt + C, () where C, C, C 3, α, and α are arbitrary constants; F (t) is an arbitrary function about t; p is an arbitrary function about y. Thus from Eqs. () and Eq. () the following general form of nonlinear wave solutions of the (+)-dimensional break soliton Eq. () is derived, u = α m α m 3 α l f (ξ) 3 α l g (η), v = (α m αm )dp (y)/dy + F (t) 3 α dp dy l f (ξ) + 3(dp /dy)αl g (η), (3) where f(ξ) and g(η) satisfy Eq. (4), ξ and η satisfy ξ = α x + p 4λα F (t)dt + C, η = α x α p + C 4λα α F (t)dt + C 3. (4) If we choose special values of l, m, n, l, m, and n in Eq. (4), families of double nonlinear wave solutions of Eq. () can be derived from general forms of solutions (5) with Eq. (6). Because the solutions obtained here are so many, we just list parts of them. Family l = k, m = k, n =, l = k, m = k, n = k, f(ξ) = sn(ξ) or cd(ξ), g(η) = cn(η), from Table. Therefore, the double Jacobi elliptic function-like solutions of (+)-dimensional break soliton equation () are derived as follows: u = α ( k ) + α ( + k ) 3 α k sn (ξ, k) + 3 α k cn (η, k),

4 84 SUN Wei-Kun, CAO Nan-Bin, and SHEN Ya-Liang Vol. 49 v = [α (k ) + α( + k )]dp /dy + F (t) 3 α dp dy k sn (ξ, k) 3(dp /dy)αk cn (η, k), (5) u = α ( k ) + α ( + k ) 3 α k cd (ξ, k) + 3 α k cn (η, k), v = [α (k ) + α ( + k )]dp /dy + F (t) 3 α dp dy k cd (ξ, k) 3(dp /dy)α k cn (η, k), (6) where ξ and η satisfy Eq. (5). When k, the double Jacobian elliptic function-like solution (6) can degenerate into double soliton-like solution where ξ and η satisfy Eq. (5). Family u = α + α 3 α tanh (ξ) + 3 α sech (η), v = (α + α)dp /dy + F (t) 3 α dp dy tanh (ξ) 3(dp /dy)α sech (η), (7) l = k, m = k, n =, l = k, m = k, n =, f(ξ) = sn(ξ) or cd(ξ), g(η) = sn(η) or cd(η), from Table. Therefore, the double Jacobi elliptic function-like solutions of (+)-dimensional break soliton equation () are derived as follows: u = (α + α )( + k ) 3 k [α sn (ξ, k) + α sn (η, k)], v = ( + k )(α α)dp /dy + F (t) 3 α dp dy k sn (ξ, k) + 3(dp /dy)αk sn (η, k), (8) u = (α + α )( + k ) 3 k [α sn (ξ, k) + α cd (η, k)], v = ( + k )(α α)dp /dy + F (t) 3 α dp dy k sn (ξ, k) + 3(dp /dy)αk cd (η, k), (9) u = (α + α )( + k ) 3 k [α cd (ξ, k) + α cd (η, k)], v = ( + k )(α α)dp /dy + F (t) 3 α dp dy k cd (ξ, k) + 3(dp /dy)αk cd (η, k), (0) where ξ and η satisfy Eq. (5). When k, the double Jacobian elliptic function-like solution (8) can degenerate into double soliton-like solution, where ξ and η satisfy Eq. (5). Family 3 u = α + α 3 α tanh (ξ) 3 α tanh (η), v = (α α)dp /dy + F (t) 3 α α dp dy tanh (ξ) + 3(dp /dy)α tanh (η), () l =, m = k, n = k, l = k, m = k, n = k, f(ξ) = dn(ξ), g(η) = cn(η). Therefore, the double Jacobi elliptic function-like solution of (+)-dimensional break soliton equation () is obtained as follows: u = α ( k ) + α (k ) + 3 α dn (ξ, k) + 3 α k cn (η, k), v = [α (k ) + α(k )]dp /dy + F (t) + 3 α dp dy dn (ξ, k) 3(dp /dy)αk cn (η, k), () where ξ and η satisfy Eq. (5).

5 No. Double Elliptic Equation Expansion Approach and Novel Solutions of (+)-Dimensional Break Soliton Equation 85 When k, the double Jacobian elliptic function solution () can degenerate into double soliton-like solution, where ξ and η satisfy Eq. (4). Family 4 u = α α + 3 α sech (ξ) + 3 α sech (η), v = (α α)dp /dy + F (t) + 3 α dp dy sech (ξ) 3(dp /dy)α sech (η), (3) l = k, m = k, n = k, l = k, m = k, n =, f(ξ) = nc(ξ), g(η) = sc(η). Therefore the double Jacobi elliptic function-like solution of (+)-dimensional break soliton equation () is derived as follows: u = α (k ) + α ( k ) 3 α ( k ) nc (ξ, k) 3 α ( k ) sc (η, k), v = [α ( k ) + α( k )]dp /dy + F (t) 3α (dp /dy)( k ) nc (ξ, k) + (3dp /dy)α( k ) sc (η, k), (4) where ξ and η satisfy Eq. (4). When k 0, the double Jacobian elliptic function solution (4) can degenerate into period wave-like solution, where ξ and η satisfy Eq. (4). Family 5 u = α + α 3 α sec (ξ) 3 α tan (η), v = (α + α )dp /dy + F (t) 3α dp /dy sec (ξ) + 3(dp /dy)α tan (η), (5) l =, m = k, n = k ( k ), l =, m = k, n = k, f(ξ) = ds(ξ), g(η) = cs(η). Therefore the double Jacobi elliptic function-like solution of (+)-dimensional break soliton equation () is derived as follows: u = α (k ) α (k ) 3 α ds (ξ, k) 3 α cs (η, k), v = [α ( k ) + α( k )]dp /dy + F (t) 3 α dp dy ds (ξ, k) + 3(dp /dy)α cs (η, k), (6) where ξ and η satisfy Eq. (4). When k 0, the double Jacobian elliptic function solution (6) can degenerate into period wave-like solution, u = α + α 3 α csc (ξ) 3 α cot (η), v = (α + α)dp /dy + F (t) 3 α dp dy csc (ξ) + 3(dp /dy)α cot (η), (7) where ξ and η satisfy Eq. (4). When k, the double Jacobian elliptic function solution (6) can degenerate into double solitary wave-like solution, u = α α 3 α csch (ξ) 3 α csch (η), v = (α α )dp /dy + F (t) 3 α dp dy csch (ξ) + 3(dp /dy)α csch (η), (8)

6 86 SUN Wei-Kun, CAO Nan-Bin, and SHEN Ya-Liang Vol. 49 where ξ and η satisfy Eq. (4). Remark In the same way, we can also obtain some other exact solutions of the (+)-dimensional break soliton equation () by properly choosing the values of l, m, n, l, m, and n. For simplicity, we cancel them here. By using of symbolic computation, all the solutions of Eq. () obtained above have been verified by putting them back into the original Eq. (). In every exact solution of Eq. () there are two different independent variables ξ and η, and the concrete expressions of ξ and η are ξ = α x + p 4λα F (t)dt + C, η = α x α p + C 4λα F (t)dt + C 3, (9) α from which we find that ξ and η satisfy relationship, η = α ξ α (p + C ) + C + C 3, p = p (y). (30) α α In addition, the independent variables ξ and η conclude two arbitrary functions F (t) and p, so these solutions are nonlinear wave solutions and they may be of important significance for the explanation of some practical physical problems. 4 Conclusion In conclusion, based on Wang s ideas [,4] the double elliptic equation expansion approach is proposed to construct some nontrivial double nonlinear wave solutions of the (+)-dimensional break soliton equation. Compared with Chen et al. s rational expansion method, [9,0] the double elliptic equation expansion approach is more powerful and effective to construct the double nonlinear wave solutions of nonlinear wave solutions such as the double Jacobian elliptic function-like solutions, the solitary wave-like solutions, and so on. In fact, the approach of this paper can also be successfully applied to some other nonlinear wave equations, such as the (+)-dimensional asymmetric version of the Nizhnik Novikov Veselov equation, the (+)-dimensional Burgers equation, etc. We always suppose that the new form of solutions obtained in this paper are not just only the extension from the mathematical meaning, but in the hope that they will lead to a deeper and more comprehensive understanding of the complex structures resulted from the nonlinearity of nonlinear wave equations. Acknowledgments The authors are very grateful to the referees for their valuable suggestions. This paper would not exist without the inspiration and helpful discussions provided by Dr. Deng-Shan Wang. The authors also thank him for his kind comments. References [] M.J. Ablowitz and P.A. Clarkson, Soliton, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge (99). [] C.S. Gardner, et al., Phys. Rev. Lett. 9 (976) 095. [3] M.R. Miura, Bäcklund Transformation, Springer-Verlag, Berlin (978). [4] D.S. Wang and H.Q. Zhang, Int. J. Mod. Phys. C 6 (005) 393. [5] R. Hirota, Phys. Rev. Lett. 7 (97) 9. [6] M.L. Wang, Phys. Lett. A 6 (996) 67. [7] W. Malfliet, Am. J. Phys. 60 (99) 659. [8] X. Tang, S.Y. Lou, and Y. Zhang, Phys. Rev. E 66 (00) 04660; S.Y. Lou, et al., Mod. Phys. Lett. B 8 (00) 075. [9] Y. Chen and Q. Wang, Commun. Theor. Phys. (Beijing, China) 45 (006) 4; Y. Chen and Q. Wang, Phys. Lett. A 347 (005) 5. [0] Y. Chen and Q. Wang, Appl. Math. Comput. 73 (006) 63; Y. Chen and Q. Wang, Appl. Math. Comput. 77 (006) 396. [] D.S. Wang and H.Q. Zhang, Z. Naturforsch 60a (005) ; D.S. Wang and H.Q. Zhang, Chaos, Solitons and Fractals 5 (005) 60; D.S. Wang, Y.F. Liu, and H.Q. Zhang, Appl. Math. Comput. 68 (005) 83. [] K.W. Chow, J. Math. Phys. 36 (995) 45. [3] Z. Yan, J. Phys. A: Math. Gen. 36 (003) 96; Z. Yan, Comput. Phys. Commun. 48 (00) 30; Z. Yan, Phys. Lett. A 33 (004) 93. [4] D.S. Wang and H. Li, Chaos, Solitons and Fractals (007), doi:0.06/j.chaos ; L. Cao, D.S. Wang, and L.X. Chen, Commun. Theor. Phys. (Beijing, China) 47 (007) 70. [5] E. Fan, Phys. Lett. A 94 (00) 6; E. Fan and Y. Hon, Chaos, Solitons and Fractals 5 (003) 559. [6] D.S. Wang, Y.J. Ren, and H.Q. Zhang, Appl. Math. E-Notes, 5 (005) 57; D.S. Wang and H. Li, Appl. Math. Comput. 88 (007) 76; D.S. Wang, H. Li, and J. Wang, Chaos, Solitons and Fractals (006), dio:0.06/j.chaos [7] Y. Chen, B. Li, and H.Q. Zhang, Commun. Theor. Phys. (Beijing, China) 40 (003) 37. [8] Y.S. Li and Y.J. Zhang, J. Phys. A: Math. Gen. 6 (993) [9] Z.Y. Yan and H.Q. Zhang, Comput. Math. Appl. 44 (00) 439. [0] O.I. Bogoyavlensky, Izv. Akad. Nauk SSSR, Ser. Mat. 53 (989) 43; 54 (990) 3; R. Radha and M. Lakshamanan, Phys. Lett. A 97 (995) 7. [] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York (99); D.V. Patrick, Elliptic Function and Elliptic Curves, Cambridge University Press, Cambridge (973).