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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 645456, 8 pages

http://dx.doi.org/10.1155/2014/645456

## Some New Exact Solutions of (1*+*2)-Dimensional Sine-Gordon Equation

Department of Physics, Zhejiang Normal University, Jinhua 321004, China

Received 31 October 2013; Revised 17 December 2013; Accepted 19 December 2013; Published 16 January 2014

Academic Editor: Baojian Hong

Copyright © 2014 Wei-Xiong Chen and Ji Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We use a generalized tanh function expansion method and a direct method to study the analytical solutions of the (1*+*2)-dimensional sine Gordon (2DsG) equation. We obtain some new interaction solutions among solitary waves and periodic waves, such as the kink-periodic wave interaction solution, two-periodic solitoff solution, and two-toothed-solitoff solution. We also investigate the propagation properties of these solutions.

#### 1. Introduction

Sine-Gordon (sG) equation is one of the most famous partial differential equations that have been investigated by many physicists for decades years. The sG equation has played a central role in lots of different scientific fields, such as in differential geometry [1], plasma physics [2], nonlinear optics [3], condensed matter physics [4], quantum field theory [5, 6], and so forth. Researchers have been spending a great deal of effort to generalize (1+1)-dimensional soliton equations to (2+1)-dimensional equations. Remarkable of these equations, in the 1980s, the Nizhnik-Novikov-Veselov (NNV) equation [7–9] and the Davey-Stewartson (DS) equation [10–12] were found. The NNV equation and the DS equation are (2+1) dimensional generalizations of the Korteweg-de Vries (KdV) equation and nonlinear Schrödinger (NLS) equation, respectively. After that, in 1991, Konopelchenko and Rogers [13, 14] proposed a significant symmetry to generalize the (1+1)-dimensional sG equation to (2+1)-dimensional sG equation through a reinterpretation and generalization of a class of infinitesimal Bäcklund transformation. The well-known nonintegrable (2+1)-dimensional sine-Gordon (2DsG) equation is as follows:

Various methods have been used to study this equation because of its rich symmetrical structure. The brief and effective methods for solving the 2DsG equation include the binary Darboux transformation [15, 16], the extensive symmetry group analysis [17, 18], Hirota’s method [19], Lamb’s method [20, 21], the Painlevé transcendents [22], and the Bäcklund transformation [23]. And researchers have found abundant types of solutions of 2DsG equation, such as the multisoliton solutions and vortex-like solution [24], line and ring solitons [25, 26], curve soliton, point instanton soliton and doubly periodic wave solutions [27–29], Solitoff structure solution, and snake-shape solitary wave solution [30].

Recently, some new useful and powerful methods have been proposed to search for the accurate solutions of nonlinear partial differential equations, such as the general algebra method for the coupled Schrödinger-Boussinesq equations [31], the general mapping deformation method for the generalized variable-coefficient Gardner equation with forcing term [32], the generalized tanh function expansion method for the Abowitz-Kaup-Nwell-Segur system [33], the bosonized supersymmetric KdV model [34], and the Broer-Kaup system [35]. Significantly, the generalized tanh function expansion method is an effective new technique for us to obtain some new interaction solutions of 2DsG equation. Also, we can solve the 2DsG equation by a direct method based on the mapping relations between 2DsG equation and the cubic nonlinear Klein-Gordon (CNKG) equation. This method can be also applied to solve the double sine-Gordon equation, the triple sine-Gordon equation, and the Ginzburg-Landau equation [36], and so forth. In this paper, we want to seek more interaction solutions of new types among solitary waves and periodic waves of the 2DsG equation by the generalized tanh function expansion method and the direct method.

This paper is organized as follows. In Section 2, a kink-periodic wave interaction solution of 2DsG equation is obtained by using of the generalized tanh function expansion method. In Section 3, two-periodic solitoff solution, periodic soliton-periodic travelling wave interaction solution, two-toothed-solitoff solution, and periodic solitoff-kink interaction solution of 2DsG equation are obtained by using the direct method. In Section 4, a short summary and discussions are given.

#### 2. Kink-Periodic Wave Interaction Solutions

The 2DsG equation (1) cannot be solved directly by the generalized tanh function expansion method [33–35], and to find some soliton-periodic wave interaction solutions of 2DsG equation, we suppose and take the following coordinates transformation: Then, we substitute (2) with (3) into (1) and arrive at with the constants , , , and satisfying It is worth noting that (4) can be solved by using the generalized tanh function expansion method. Firstly, we set where , , , and are functions of variables . In order to obtain some soliton-periodic wave interaction solutions, let where , , in which , , , and are undetermined constants. Then, we substitute (6) and (7) into (4) and analyse the coefficients of function order by order we get the expression of where the functions and satisfy Furthermore, is a solution of the following Jacobi elliptic function equation: with these parameters , , , and satisfying where , , , and , in which and is a constant.

Now we choose the sine Jacobi elliptic function as a solution of (10), and the functions and are easy to be obtained Then substituting (12), (13), and (11) into (10), relationships of these parameters are written as where is the modulus of the Jacobi elliptic function .

Finally, the accurate expression of is gained: where , , and . The solution of (15) denotes a kink-periodic wave interaction solution of 2DsG equation. Velocities of these two travelling waves are and , respectively.

Figure 1 shows the density distribution of a kink-periodic wave interaction solution on the *x*-*y* plane given by and (15) with these parameters
at time . This figure exhibits a special interaction structure of a kink and a periodic wave. Figure 2 shows the propagation of the kink-periodic wave solution at and . In this figure, the soliton propagates along the negative direction of the *x*-axis, and its velocity is quicker than the one of the periodic wave, which also propagates along the negative direction.

#### 3. Solitoff, Periodic Soliton-Periodic Travelling Wave, and Periodic Solitoff-Kink Interaction Solutions

In this section, we use the direct method to study the 2DsG equation. Based on the Lamb substitution [20, 21], the solution of (1) can be set to the following form: in which the function is the solution of the CNKG equation [30, 36], under the constrained condition with . Function can be various styles, such as , , , and [36]. Here we take where function , in which is a function of variables , and the constant is the modulus of the Jacobi elliptic function. Then, we substitute (17) and (20) into 2DsG equation and get with the constrained conditions Here we define then an arbitrary function can be included in the function by solving (22), namely, and parameters , , , , , and satisfy where the sign “” in (22) and (25) takes “−” when and takes “+” when . Due to the existence of the arbitrary functions, abundant exact solutions of (1) will be obtained as long as the function is properly selected.

When we take a (2+1)-dimensional two-periodic solitoff solution of 2DsG equation can be obtained: We know that a solitoff is defined as a half line soliton. The solution of (27) indicates a solitoff type solution constructed by two travelling waves that propagate in different directions. Velocities of these two travelling waves are and , respectively.

Figure 3 shows a two-periodic solitoff solution (27) with these parameters
at time . The angle of the two-periodic solitoff in this figure is actually an obtuse angle although it seems to be orthogonal. It is because . Figure 4 shows more details of the two-periodic solitoff solution (27) with (28). The two-periodic solitoff solution with different wavelength has the same amplitude and keeps the peak unchanged during the propagation process. Their phase velocities are different, but their travelling directions are same; they propagate along the negative *y*-axis.

A periodic soliton-periodic travelling wave interaction solution of 2DsG equation can be obtained: by choosing

Figure 5(a) shows the periodic soliton-periodic travelling wave interaction solution (29) with these parameters
at time . The solitoff-type structure solution does not appear, whereas these two travelling waves propagate in the different directions. The graph is similar to the soliton-periodic interaction wave in [33], but the soliton really has the periodicity and the peak of the soliton keeps periodically changing. Figure 5(b) shows the density distribution of on the *x*-*y* plane.

Furthermore, if we take then a two-sawtooth-solitoff solution and a periodic solitoff-kink interaction solution of 2DsG equation can be written asrespectively. Figure 6 shows a two-toothed-solitoff solution (33) with (31) in the limit case of the modulus . The two-toothed-solitoff structure is constructed by a kink soliton and an antikink soliton. Their travelling velocities are different, but group velocities are the same. And travelling directions of these two solitoff waves construct a constant acute angle during the propagation process.

Figure 7(a) displays a periodic solitoff-kink interaction solution constructed by a bright soliton and a kink soliton. Figures 7(b)–7(d) show that the bright soliton and the kink soliton have different travelling velocities, and they propagate along the negative *x*-axis. The peak of the bright soliton keeps increasing until it is arriving at the same amplitude of the kink soliton.

#### 4. Summary and Discussion

First of all, we use the generalized tanh function expansion method to solve the 2DsG equation; a special new kink-periodic wave interaction solution is explicitly expressed both analytically and graphically. This interaction solution between tanh-type soliton and periodic wave of 2DsG equation is firstly obtained. Then, we use the direct method and obtain more new interaction solutions of the 2DsG equation, including the two-periodic solitoff solution (27), periodic soliton-periodic travelling wave interaction solution (29), two-toothed-solitoff solution (33), and periodic solitoff-kink interaction solution (34). The solution (34) is a generalization of a single straight-line kink soliton solution, while the solution (33) is an alternative generalization of periodic straight-line solitoff type of kink soliton solution. These types of interaction solutions are also firstly found for the 2DsG equation. All of these solutions indicate the interaction solution among solitary waves and periodic waves; their travelling velocities are different, but group velocities are same, and they propagate in different trajectories which contain linear shape, curve shape, and saw-tooth shape. In fact, the forms of (12) and (20) can be not only taken the sine Jacobi elliptic function , more functions can be selected such as , , and , and more explicit solutions can be gained. The abundant solutions solved by these two methods suggest that the rich structures of nonlinear systems do not only exist in the integrable systems but also in the nonintegrable systems. Furthermore, there are some types of localized solutions decaying in all directions, for instance, the dromions and ring solitons have not been found by these two methods; those will be left for us to do more research.

#### Conflict of Interests

The authors declare that they have no financial relationships with other people or organizations that can inappropriately influence this work or possible conflict of interests.

#### Acknowledgments

The work was supported by the National Natural Science Foundation of China no. 11175158 and by program for Innovative Research Team in Zhejiang Normal University.

#### References

- L. P. Eisenhart,
*A Treatise on the Differential Geometry of Curves and Surfaces*, Dover Publications, New York, NY, USA, 1960. View at MathSciNet - H. Washimi and T. Taniuti, “Propagation of ion-acoustic solitary waves of small amplitude,”
*Physical Review Letters*, vol. 17, no. 19, pp. 996–998, 1966. View at Publisher · View at Google Scholar · View at Scopus - H. Leblond and D. Mihalache, “Ultrashort light bullets described by the two-dimensional sine-Gordon equation,”
*Physical Review A*, vol. 81, no. 6, Article ID 063815, 2010. View at Publisher · View at Google Scholar · View at Scopus - I. Loutsenko and D. Roubtsov, “Critical velocities in exciton superfluidity,”
*Physical Review Letters*, vol. 78, no. 15, pp. 3011–3014, 1997. View at Publisher · View at Google Scholar · View at Scopus - S. Samuel, “Grand partition function in field theory with applications to sine-Gordon field theory,”
*Physical Review D*, vol. 18, no. 6, pp. 1916–1932, 1978. View at Publisher · View at Google Scholar · View at Scopus - A. Barone, F. Esposito, C. J. Magee, and A. C. Scott, “Theory and applications of the sine-gordon equation,”
*La Rivista del Nuovo Cimento*, vol. 1, no. 2, pp. 227–267, 1971. View at Publisher · View at Google Scholar · View at Scopus - S. P. Novikov and A. P. Veselov, “Two-dimensional Schrödinger operator: inverse scattering transform and evolutional equations,”
*Physica D*, vol. 18, no. 1–3, pp. 267–273, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Athorne and J. J. C. Nimmo, “On the Moutard transformation for integrable partial differential equations,”
*Inverse Problems*, vol. 7, no. 6, article 809, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Radha and M. Lakshmanan, “Singularity analysis and localized coherent structures in $(2+1)$-dimensional generalized Korteweg-de Vries equations,”
*Journal of Mathematical Physics*, vol. 35, no. 9, pp. 4746–4756, 1994. View at Publisher · View at Google Scholar · View at MathSciNet - M. Boiti, J. Jp. Leon, L. Martina, and F. Pempinelli, “Scattering of localized solitons in the plane,”
*Physics Letters A*, vol. 132, no. 8-9, pp. 432–439, 1988. View at Publisher · View at Google Scholar · View at MathSciNet - A. S. Fokas and P. M. Santini, “Dromions and a boundary value problem for the Davey-Stewartson $(2+1)$ equation,”
*Physica D*, vol. 44, no. 1-2, pp. 99–130, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Davey and K. Stewartson, “On three-dimensional packets of surface waves,”
*Proceedings of the Royal Society A*, vol. 338, pp. 101–110, 1974. - B. G. Konopelchenko and C. Rogers, “On $(2+1)$-dimensional nonlinear systems of Loewner-type,”
*Physics Letters A*, vol. 158, no. 8, pp. 391–397, 1991. View at Publisher · View at Google Scholar · View at MathSciNet - B. G. Konopelchenko and C. Rogers, “On generalized Loewner systems: novel integrable equations in $2+1$ dimensions,”
*Journal of Mathematical Physics*, vol. 34, no. 1, pp. 214–242, 1993. View at Publisher · View at Google Scholar · View at MathSciNet - W. K. Schief, “On the geometry of an integrable $(2+1)$-dimensional sine-Gordon system,”
*Proceedings of the Royal Society A*, vol. 453, no. 1963, pp. 1671–1688, 1997. View at Publisher · View at Google Scholar · View at MathSciNet - W. K. Schief, “On localized solitonic solutions of a $(2+1)$-dimensional sine-Gordon system,”
*Journal of Physics A*, vol. 25, no. 24, pp. L1351–L1354, 1992. View at Publisher · View at Google Scholar · View at MathSciNet - P. A. Clarkson, E. L. Mansfield, and A. E. Milne, “Symmetries and exact solutions of a $(2+1)$-dimensional sine-Gordon system,”
*Philosophical Transactions of the Royal Society A*, vol. 354, no. 1713, pp. 1807–1835, 1996. View at Publisher · View at Google Scholar · View at MathSciNet - S.-y. Lou, “Symmetry analysis and exact solutions of the $(2+1)$ dimensional sine-Gordon system,”
*Journal of Mathematical Physics*, vol. 41, no. 9, pp. 6509–6524, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Hirota, “Exact three-soliton solution of the two-dimensional sine-Gordon equation,”
*Journal of the Physical Society of Japan*, vol. 35, no. 5, p. 1566, 1973. View at Publisher · View at Google Scholar · View at Scopus - J. Zagrodziński, “Particular solutions of the sine-Gordon equation in $2+1$ dimensions,”
*Physics Letters A*, vol. 72, no. 4-5, pp. 284–286, 1979. View at Publisher · View at Google Scholar · View at MathSciNet - P. L. Christiansen and O. H. Olsen, “Return effect for rotationally symmetric solitary wave solutions to the sine-Gordon equation,”
*Physics Letters A*, vol. 68, no. 2, pp. 185–188, 1978. View at Publisher · View at Google Scholar · View at MathSciNet - P. Kaliappan and M. Lakshmanan, “Kadomtsev-Petviashvile and two-dimensional sine-Gordon equations: reduction to Painlevé transcendents,”
*Journal of Physics A*, vol. 12, no. 10, pp. L249–L252, 1979. View at Publisher · View at Google Scholar · View at MathSciNet - B. G. Konopelchenko, W. Schief, and C. Rogers, “A $(2+1)$-dimensional sine-Gordon system: its auto-Bäcklund transformation,”
*Physics Letters A*, vol. 172, no. 1-2, pp. 39–48, 1992. View at Publisher · View at Google Scholar · View at MathSciNet - S. Takeno, “Multi-(resonant-soliton)-soliton solutions and vortex-like solutions to two- and three-dimensional sine-Gordon equations,”
*Progress of Theoretical Physics*, vol. 68, no. 3, pp. 992–995, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Djidjeli, W. G. Price, and E. H. Twizell, “Numerical solutions of a damped sine-Gordon equation in two space variables,”
*Journal of Engineering Mathematics*, vol. 29, no. 4, pp. 347–369, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Dehghan and D. Mirzaei, “The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation,”
*Computer Methods in Applied Mechanics and Engineering*, vol. 197, no. 6-8, pp. 476–486, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S.-y. Lou, “Symmetry analysis and exact solutions of the $2+1$ dimensional sine-Gordon system,”
*Journal of Mathematical Physics*, vol. 41, no. 9, pp. 6509–6524, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. W. Chow, “A class of doubly periodic waves for nonlinear evolution equations,”
*Wave Motion*, vol. 35, no. 1, pp. 71–90, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X.-Y. Tang, S.-Y. Lou, and Y. Zhang, “Localized excitations in $(2+1)$-dimensional systems,”
*Physical Review E*, vol. 66, no. 4, Article ID 046601, p. 17, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - S. Y. Lou, H.-C. Hu, and X.-Y. Tang, “Interactions among periodic waves and solitary waves of the $(N+1)$-dimensional sine-Gordon field,”
*Physical Review E*, vol. 71, no. 3, Article ID 036604, 2005. View at Publisher · View at Google Scholar · View at Scopus - B. Hong and D. Lu, “New exact Jacobi elliptic function solutions for the coupled Schrödinger-Boussinesq equations,”
*Journal of Applied Mathematics*, vol. 2013, Article ID 170835, 7 pages, 2013. View at Publisher · View at Google Scholar - B. Hong and D. Lu, “New exact solutions for the generalized variable-coefficient Gardner equation with forcing term,”
*Applied Mathematics and Computation*, vol. 219, no. 5, pp. 2732–2738, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - S. Y. Lou, X. P. Cheng, and X. Y. Tang, “Interactions between solitons and other nonlinear Schrödinger waves,” http://arxiv.org/abs/1208.5314.
- X. N. Gao, S. Y. Lou, and X. Y. Tang, “Bosonization, singularity analysis, nonlocal symmetry reductions and exact solutions of supersymmetric KdV equation,”
*Journal of High Energy Physics*, vol. 5, no. 29, 2013. View at MathSciNet - C. L. Chen and S. Y. Lou, “CTE solvability and exact solution to the Broer-Kaup system,”
*Chinese Physics Letters*, vol. 30, Article ID 110202, pp. 2–4, 2013. - S. Y. Lou and G. J. Ni, “The relations among a special type of solutions in some $(D+1)$-dimensional nonlinear equations,”
*Journal of Mathematical Physics*, vol. 30, no. 7, pp. 1614–1620, 1989. View at Publisher · View at Google Scholar · View at MathSciNet