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Double Periodic Wave Solutions of the (2 + 1)-Dimensional Sawada-Kotera Equation
Based on a general Riemann theta function and Hirota’s bilinear forms, we devise a straightforward way to explicitly construct double periodic wave solution of -dimensional nonlinear partial differential equation. The resulting theory is applied to the -dimensional Sawada-Kotera equation, thereby yielding its double periodic wave solutions. The relations between the periodic wave solutions and soliton solutions are rigorously established by a limiting procedure.
It is always important to investigate the exact solutions for nonlinear evolution equations, which play an important role in the study of nonlinear models of natural and social phenomena. Nonlinear wave phenomena appears in various scientific and engineering areas, such as fluid mechanics, theory of solitons, hydrodynamics, and theory of turbulence, optical fibers, chaos theory, biology, and chemical physics. In the last three decades, various powerful methods have been presented, such as extended tanh method , homogeneous balance method , Lie group method , Wronskian technique [4, 5], Darboux transformation method , Hirota’s bilinear method [7–10], and algebro-geometrical approach .
The Hirota’s bilinear method provides a powerful way to derive soliton solutions to nonlinear integrable equations and its basis is the Hirota bilinear formulation. Once the corresponding bilinear forms are obtained, multisoliton solutions and rational solutions to nonlinear differential equations can be computed in quite a systematic way. In 1980s, based on Hirota bilinear forms, Nakamura proposed a comprehensive method to construct a kind of multiperiodic solutions of nonlinear equations in his papers [12, 13], such a method of solution does not need any Lax pairs and their induced Riemann surfaces for the considered equations. The advantage of the method is that it only relies on the existed Hirota bilinear forms. Moreover, all parameters appearing in Riemann matrices are completely arbitrary, whereas algebro-geometric solutions involve specific Riemann constants, which are usually difficult to compute.
In recent years, Hon et al. have extended this method to investigate the discrete Toda lattice , -dimensional modified Bogoyavlenskii-Schiff equation , and the Supersymmetric KdV-Sawada-Kotera-Ramani equation . Ma et al. constructed one-periodic and two-periodic wave solutions to a class of -dimensional Hirota bilinear equations . Tian and Zhang gave the exact periodic solutions for some evolution equations with the aid of the Hirota bilinear method and theta functions identities [18, 19].
Our aim in the present work is to improve the main steps of the existing methods of Fan and Chow in  into the case of three dimensions. We propose a theorem, which actually provides us a direct and unifying way for applying in a class of -dimensional nonlinear partial differential equations. Once such an equation is written in a bilinear form, its double periodic wave solutions can be directly obtained by using this theorem.
The organization of this paper is as follows. In Section 2, we briefly introduce the Hirota bilinear operator and the Riemann theta function. In particular, we present a theorem for constructing periodic wave solutions for -dimensional nonlinear partial differential equations. As applications of our method, in Section 3, we construct double periodic wave solutions to the -dimensional Sawada-Kotera equation. In addition, it is rigorously shown that the double periodic wave solutions tend to the soliton solutions under small amplitude limits. Finally, some conclusions and discussions are presented in Section 4.
2. Hirota Bilinear Operator and Riemann Theta Function
In this section we briefly present the notation that will be used in this paper. Here the bilinear operators are defined by with .
Proposition 1. The Hirota bilinear operators have properties
where , with being constants. More generally, one has
where is a polynomial about operators . These properties are useful in deriving Hirota’s bilinear form and constructing periodic wave solutions of nonlinear equations.
Then, one would like to consider a general Riemann theta function and discuss its periodicity; the Riemann theta function reads where , complex parameter , , and complex phase variables , which is called the period matrix of the Riemann theta function.
In the definition of the theta function (4), for the case , hereafter, one uses the notation of , for simplicity. Moreover, one has .
Definition 2. A function on is said to be quasiperiodic in with fundamental periods if being linearly dependent over and there exists a function in , such that
In particular, is called double periodic as , and it becomes periodic with the period if and only if .
Proposition 3. The theta function has the periodic properties: One regards the vectors 1 and as periods of the theta function with multipliers 1 and , respectively.
Proposition 4. The meromorphic functions on is as follows: then it holds that that is, is a double periodic function with 1 and .
Theorem 5. Assuming that and are two Riemann theta functions with , then Hirota bilinear operators , , and exhibit the following perfect properties when they act on a pair of theta functions:
where the notation represents two different transformations corresponding to . The bilinear formulas for , are the same as (12) by replacing with and .
In general, for a polynomial operator with respect to , , and , one has the following useful formula: where and one denotes vector .
Proof. Making use of the formula (2), we obtain the relation Shifting summation index as , then Formula (13) follows from (12). Formulas (13) and (14) imply that if the following equations are satisfied then and are periodic wave solutions of the bilinear equation:
Remark 6. Formula (17) actually provides us an unified approach to construct periodic wave solutions for nonlinear equations. Once an equation is written bilinear forms, then its periodic wave solutions can be directly obtained by solving system (17).
3. The (2 + 1)-Dimensional Sawada-Kotera Equation
In this section, we will focus on the following -dimensional Sawada-Kotera ()DSK) model [20–22]: where is a function of the variables , and , and the other quantities are similarly defined. It was widely used in many branches of physics, such as conformal field theory, two-dimensional quantum gravity gauge field, theory, and nonlinear science Liuvill flow conservation equations. When , (19) reduces to the Sawada-Kotera equation : Equation (19), a B-type Kadomtsev-Petviashvili (KP) model, has also been referred to BKP equation because it is associated with a B-type group . Through the truncated Painlevé expansion and Hirota bilinear method, multisoliton solutions of (19) have been derived and graphically discussed in . In the framework of Bell-polynomial manipulations, the Bell-polynomial expression and Bell-polynomial-typed BT for (19) have been given in . Here we construct its double periodic wave solution and show that the one-soliton solution can be obtained as limiting case of the double periodic wave solution.
3.1. Construct Double Periodic Wave Solutions of the (2 + 1)DSK Equation
Remark 7. The constant may be taken to be zero in the construction of soliton solutions. However, in our double periodic wave case, the nonzero constant plays an important role and cannot be dropped.
When , (19) admits a one-soliton solution  where phase variable , and , , are constants. Next, we turn to see the periodicity of the solution (23); the function is chosen to be a Riemann theta function; namely, where phase variable . According to Proposition 4, we refer to which shows that the solution is a double periodic function with two fundamental periods 1 and .
By introducing the notations as Substituting (24) into (22), using formulas (17) and (26), leads to the following linear system: where we have denoted by the notations
This system admits an explicit solution where we have omitted the notation after , for simplicity of formula (29). Therefore, we get a double periodic wave solution of (19) which reads with the theta function given by (4) for the case , and parameters , by (29), while other parameters , , are free.
3.2. Feature and Asymptotic Property of Double Periodic Waves
The double periodic wave solution (30) possesses a simple characterization as follows.(i)It has a single phase variable ; that is, it is one-dimensional.(ii)It has two fundamental periods 1 and in the phase variable .(iii)The speed parameter of is given by (iv)It has only one wave pattern for all time and it can be viewed as a parallel superposition of overlapping one-soliton waves, placed one period apart.
Now, we further consider the asymptotic properties of the double periodic wave solution. The relation between the periodic wave solution (30) and the one-soliton solution (23) can be established as follows.
It implies that the double periodic solution (30) tends to the one-soliton solution (23) under a small amplitude limit. In other words, the periodic solution (30) tends to a solution under a small amplitude limit; namely
Proof. We explicitly expand the coefficients of system (27) as follows: Let the solution of the system (27) be in the form Substituting the expansions (35) and (36) into the system (27), where the second equation is divided by , and letting , we immediately obtain the following relations: which have a solution Combining (32) and (38) leads to Hence we conclude that In the following, we consider asymptotic properties of the double periodic wave solution (30) under the limit . For this purpose, we expand the Riemann theta function and make use of the expression (40); it follows that Therefore we conclude that the periodic solution (30) just goes to the soliton solution (23) as the amplitude .
In this paper, based on the Hirota’s bilinear method, combining the theory of a general Riemann theta function, we have derived a method of constructing double periodic wave solutions for -dimensional nonlinear partial differential equations. As application of our method, we construct double periodic wave solutions to the -dimensional Sawada-Kotera equation. The double periodic wave solutions obtained in this paper are theta function series solutions. By making a limiting procedure, we have analyzed asymptotic behavior of the double periodic waves, obtaining the relations between the periodic wave solutions and soliton solutions. We note that this method can be generalized to the case of to construct -periodic wave solutions. But more constraint equations need to be satisfied, so the calculation will be more complicated.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the Fundamental Research Funds for the Central Universities (2013XK03), the National Natural Science Foundation of China (Grant no. 11371361), and the National Natural Science Foundation of China (Grant no. 11271008).
- E. Fan, “Extended tanh-function method and its applications to nonlinear equations,” Physics Letters A, vol. 277, no. 4-5, pp. 212–218, 2000.
- M. L. Wang, “Exact solutions for a compound KdV-Burgers equation,” Physics Letters A, vol. 213, no. 5-6, pp. 279–287, 1996.
- G. Bluman and S. Kumei, Symmetries and Differential Equations, vol. 81 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1989.
- W.-X. Ma, C.-X. Li, and J. He, “A second Wronskian formulation of the Boussinesq equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 12, pp. 4245–4258, 2009.
- Y. Zhang, T.-F. Cheng, D.-J. Ding, and X.-L. Dang, “Wronskian and Grammian solutions for -dimensional soliton equation,” Communications in Theoretical Physics, vol. 55, no. 1, pp. 20–24, 2011.
- V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, Germany, 1991.
- R. Hirota, “Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons,” Physical Review Letters, vol. 27, no. 18, pp. 1192–1194, 1971.
- R. Hirota and J. Satsuma, “-soliton solutions of model equations for shallow water waves,” Journal of the Physical Society of Japan, vol. 40, no. 2, pp. 611–612, 1976.
- X.-B. Hu and W.-X. Ma, “Application of Hirota's bilinear formalism to the Toeplitz lattice—some special soliton-like solutions,” Physics Letters A, vol. 293, no. 3-4, pp. 161–165, 2002.
- X.-B. Hu and W.-M. Xue, “A bilinear Bäcklund transformation and nonlinear superposition formula for the negative Volterra hierarchy,” Journal of the Physical Society of Japan, vol. 72, no. 12, pp. 3075–3078, 2003.
- S. P. Novikov, “The periodic problem for the Korteweg-de Vries equation,” Functional Analysis and Its Applications, vol. 8, no. 3, pp. 236–246, 1974.
- A. Nakamura, “A direct method of calculating periodic wave solutions to nonlinear evolution equations. I. Exact two-periodic wave solution,” Journal of the Physical Society of Japan, vol. 47, no. 5, pp. 1701–1705, 1979.
- A. Nakamura, “A direct method of calculating periodic wave solutions to nonlinear evolution equations. II. Exact one-periodic and two-periodic wave solution of the coupled bilinear equations,” Journal of the Physical Society of Japan, vol. 48, no. 4, pp. 1365–1370, 1980.
- Y. C. Hon, E. Fan, and Z. Qin, “A kind of explicit quasi-periodic solution and its limit for the Toda lattice equation,” Modern Physics Letters B, vol. 22, no. 8, pp. 547–553, 2008.
- E. Fan and K. W. Chow, “On the periodic solutions for both nonlinear differential and difference equations: a unified approach,” Physics Letters A, vol. 374, no. 35, pp. 3629–3634, 2010.
- E. Fan, “Supersymmetric KdV-Sawada-Kotera-Ramani equation and its quasi-periodic wave solutions,” Physics Letters A, vol. 374, no. 5, pp. 744–749, 2010.
- W.-X. Ma, R. Zhou, and L. Gao, “Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in dimensions,” Modern Physics Letters A, vol. 24, no. 21, pp. 1677–1688, 2009.
- S.-F. Tian and H.-Q. Zhang, “Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations,” Journal of Mathematical Analysis and Applications, vol. 371, no. 2, pp. 585–608, 2010.
- S.-F. Tian and H.-Q. Zhang, “Riemann theta functions periodic wave solutions and rational characteristics for the -dimensional and -dimensional Ito equation,” Chaos, Solitons & Fractals, vol. 47, pp. 27–41, 2013.
- C.-W. Cao and X. Yang, “Algebraic-geometric solution to -dimensional Sawada-Kotera equation,” Communications in Theoretical Physics, vol. 49, no. 1, pp. 31–36, 2008.
- A.-M. Wazwaz, “Multiple soliton solutions for -dimensional Sawada-Kotera and Caudrey-Dodd-Gibbon equations,” Mathematical Methods in the Applied Sciences, vol. 34, no. 13, pp. 1580–1586, 2011.
- V. G. Dubrovsky and Y. V. Lisitsyn, “The construction of exact solutions of two-dimensional integrable generalizations of Kaup-Kuperschmidt and Sawada-Kotera equations via -dressing method,” Physics Letters A, vol. 295, no. 4, pp. 198–207, 2002.
- R. A. Zait, “Bäcklund transformations, cnoidal wave and travelling wave solutions of the SK and KK equations,” Chaos, Solitons & Fractals, vol. 15, no. 4, pp. 673–678, 2003.
- R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, UK, 2004.
- X. Lü, T. Geng, C. Zhang, H.-W. Zhu, X.-H. Meng, and B. Tian, “Multi-soliton solutions and their interactions for the -dimensional Sawada-Kotera model with truncated Painlevé expansion, Hirota bilinear method and symbolic computation,” International Journal of Modern Physics B, vol. 23, no. 25, pp. 5003–5015, 2009.
- X. Lü, B. Tian, K. Sun, and P. Wang, “Bell-polynomial manipulations on the Bäcklund transformations and Lax pairs for some soliton equations with one Tau-function,” Journal of Mathematical Physics, vol. 51, no. 11, Article ID 113506, 2010.
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