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

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: 𝜕²Φ𝜕𝑥²+𝜕²Φ𝜕𝑦²−𝜕²Φ𝜕𝑡²=𝑚sinΦ. (1)

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 Φ=−𝑖ln[𝑊(𝑋,𝑇)], (2) and take the following coordinates transformation: 𝑋=𝑥+𝛼₁𝑦+𝛽₁𝑡,𝑇=𝛼₂𝑥+𝑦+𝛽₂𝑡. (3) Then, we substitute (2) with (3) into (1) and arrive at 𝑊_𝑋𝑊_𝑇−𝑊𝑊_𝑋𝑇+𝑊−𝑊³=0, (4) with the constants 𝛼₁, 𝛼₂, 𝛽₁, and 𝛽₂ satisfying 𝛼²₂=𝛽²₂−1,𝛽²₁=1+𝛼²₁,𝛼₁=𝛽₁𝛽₂−𝛼₂+𝑚4. (5) It is worth noting that (4) can be solved by using the generalized tanh function expansion method. Firstly, we set 𝑊=𝑢₂tanh²(Ψ)+𝑢₁tanh(Ψ)+𝑢₀, (6) where 𝑢₂, 𝑢₁, 𝑢₀, and Ψ are functions of variables (𝑋,𝑇). In order to obtain some soliton-periodic wave interaction solutions, let Ψ=𝜉₁+𝜓₁(𝜉), (7) where 𝜉₁=𝑘₁𝑋+𝜔₁𝑇, 𝜉=𝑘𝑋+𝜔𝑇, in which 𝑘, 𝜔, 𝑘₁, and 𝜔₁ are undetermined constants. Then, we substitute (6) and (7) into (4) and analyse the coefficients of function tanh(Ψ) order by order we get the expression of 𝑊𝑊=2𝜔₁+𝜓₂(𝜉)𝜔𝑘₁+𝜓₂(𝜉)𝑘tanh²(Ψ)−2𝜓₃(𝜉)(Ψ)+𝜓𝑘𝜔tanh₂₃(𝜉)𝑘²𝜔²2𝜔₁+𝜓₂(𝜉)𝜔𝑘₁+𝜓₂(𝜉)𝑘, (8) where the functions 𝜓₂(𝜉) and 𝜓₃(𝜉) satisfy 𝜓₂(𝜉)=𝜓_1𝜉(𝜉),𝜓₃(𝜉)=𝜓_2𝜉(𝜉). (9) Furthermore, 𝜓₂(𝜉) is a solution of the following Jacobi elliptic function equation: 𝜓²_2𝜉(𝜉)=𝑎₀+𝑎₁𝜓₂(𝜉)+𝑎₂𝜓²₂(𝜉)+𝑎₃𝜓³₂(𝜉)+4𝜓⁴₂(𝜉), (10) with these parameters 𝑎₀, 𝑎₁, 𝑎₂, and 𝑎₃ satisfying 𝑎₀=2𝜔₁𝑘₁2𝛾𝑘₁𝜔₁𝑔+𝛾²𝑘₁𝑘−𝜔₁𝜔𝑘²𝜔²,𝑎𝛾𝑔₁=24𝛾𝑘₁𝜔₁𝑔ℎ+𝛾²𝑘₁𝑘𝑞−𝜔₁𝜔𝑝𝑘²𝜔²,𝑎𝛾𝑔₂=24𝛾𝑘₁𝑘𝜔₁𝜔𝑔+𝛾²𝑘²𝑝+2𝛾𝑔ℎ²−𝜔²𝑞𝑘²𝜔²,𝑎𝛾𝑔₃=2𝛾²𝑘²+4𝛾𝑔ℎ−𝜔²,𝑘𝜔𝛾𝑔 (11) where 𝑔=𝑘𝜔₁−𝑘₁𝜔, ℎ=𝑘𝜔₁+𝑘₁𝜔, 𝑝=(3ℎ−𝑔)/2, and 𝑞=(3ℎ+𝑔)/2, in which 𝑘𝜔₁≠𝑘₁𝜔 and 𝛾 is a constant.

Now we choose the sine Jacobi elliptic function as a solution of (10), 𝜓₂(𝜉)=𝑎sn(𝑏𝜉), (12) and the functions 𝜓₁(𝜉) and 𝜓₃(𝜉) are easy to be obtained 𝜓₁(𝜉)=𝑎lndn(𝑏𝜉)−𝑛cn(𝑏𝜉),𝜓𝑏𝑛₃(𝜉)=𝑎𝑏cn(𝑏𝜉)dn(𝑏𝜉). (13) Then substituting (12), (13), and (11) into (10), relationships of these parameters are written as 𝑏=2𝑎𝑛,𝜔₁𝑘=−¹𝜔𝛾𝑘²𝛾𝑛²−4𝑎²𝜔²𝑛²+1+𝜔²𝑛²8𝛾𝑘²₁+1𝑘𝛾𝑘²𝛾𝑛²+4𝑎²𝜔²𝑛²+1−𝜔²𝑛²8𝛾𝑘²₁,+3𝜔=𝑛𝑘𝛾−21+8𝛾𝑘²₁2𝑎²𝑘²𝛾𝑛²+1−𝑛²4𝛾𝑘²₁+1+𝑛²,𝑛𝛾=−−𝑘₁𝑛+𝑎²𝑘²𝑛²+1−𝑘²₁𝑛²4𝑘₁𝑎²𝑘²𝑛²+1−2𝑘²₁𝑛²,𝑘₁=√2𝑘𝑎²𝑛²+1±𝑎𝑎²𝑛²+1²−𝑏²𝑛⁴,2𝑛 (14) where 𝑛 is the modulus of the Jacobi elliptic function sn(𝑧)=sn(𝑧,𝑛).

Finally, the accurate expression of Φ is gained: Φ=−𝑖ln2𝜔₁+𝑎𝜔𝑆𝑘₁𝜉+𝑎𝑘𝑆×tanh₁+𝑎ln(𝐷−𝑛𝐶)𝑏𝑛−𝑎𝑏𝑘𝜔𝐶𝐷²×2𝜔₁+𝑎𝜔𝑆𝑘₁+𝑎𝑘𝑆⁻¹, (15) where 𝑆=sn(𝑏𝜉), 𝐶=cn(𝑏𝜉), and 𝐷=dn(𝑏𝜉). The solution of (15) denotes a kink-periodic wave interaction solution of 2DsG equation. Velocities of these two travelling waves are 𝑣₁=(𝑘₁𝛽₁+𝜔₁𝛽₂)/[(𝑘₁+𝜔₁𝛼₂)²+(𝑘₁𝛼₁+𝜔₁)²]^1/2 and 𝑣₂=(𝑘𝛽₁+𝜔𝛽₂)/[(𝑘+𝜔𝛼₂)²+(𝑘𝛼₁+𝜔)²]^1/2, respectively.

Figure 1 shows the density distribution of a kink-periodic wave interaction solution on the x-y plane given by [−exp(𝑖Φ)] and (15) with these parameters 4𝑎=59,𝑛=710,𝑘=2,𝑏=169,𝑘₁=289,√𝜔=−135108512,𝜔₁√=−271012128,𝑚=4,𝛼₁𝛼=−2,₂√=−33+51364,𝛽₁=√5,𝛽₂√=−335+√65.64 (16) at time 𝑡=1. 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 𝑦=0 and 𝑡=1. 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 𝑥 direction.

Figure 1 

The density distribution of a kink-periodic wave interaction solution [−exp(𝑖Φ)] and (15) with (16) on the x-y plane at time 𝑡=1.

Figure 2 

The propagation of the kink-periodic interaction solution [−exp(𝑖Φ)] and (15) with (16) at 𝑦=0 and 𝑡=1.

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: Φ𝑥,𝑦,𝑡=4arctan𝑀𝑥,𝑦,𝑡, (17) in which the function 𝑀(𝑥,𝑦,𝑡) is the solution of the CNKG equation [30, 36], 𝑀_𝑥𝑥+𝑀_𝑦𝑦−𝑀_𝑡𝑡=𝜆𝑀+𝜇𝑀³, (18) under the constrained condition 𝑀²_𝑥+𝑀²_𝑦−𝑀²_𝑡=𝜆𝑀²+𝜇2𝑀⁴+𝜇2, (19) with 𝑚=𝜆−𝜇. Function 𝑀(𝑥,𝑦,𝑡) can be various styles, such as exp, tanh, sn, and dn [36]. Here we take 𝑀=√𝑛sn𝑉, (20) where function 𝑉=(|𝑚|𝑉)/(𝑛+1), in which 𝑉 is a function of variables (𝑥,𝑦,𝑡), and the constant 𝑛 is the modulus of the Jacobi elliptic function. Then, we substitute (17) and (20) into 2DsG equation and get |𝑚|∇𝑉²+𝑚|𝑚|𝑛²+𝑛sn³𝑉+𝑛|𝑚|□𝑉dn𝑉cn𝑉sn²𝑉−|𝑚|∇𝑉²+𝑚|𝑚|()𝑉+√𝑛+1sn𝑛|𝑚|□𝑉dn𝑉cn𝑉=0, (21) with the constrained conditions □𝑉=0,∇𝑉²=±1. (22) Here we define □=𝜕²_𝑥+𝜕²_𝑦−𝜕²_𝑡,∇²=𝜕_𝑥²+𝜕_𝑦²−𝜕_𝑡²; (23) then an arbitrary function 𝑣(𝜉) can be included in the function 𝑉 by solving (22), namely, 𝑉=𝑣(𝜉)+𝜉₀=𝑣𝑘₁₁𝑥+𝑘₁₂𝑦+𝜔₁𝑡+𝑘₀₁𝑥+𝑘₀₂𝑦+𝜔₀𝑡, (24) and parameters 𝑘₀₁, 𝑘₀₂, 𝑘₁₁, 𝑘₁₂, 𝜔₀, and 𝜔₁ satisfy 𝑘²₀₁+𝑘²₀₂−𝜔²₀𝑘=±1,₂₁₁+𝑘²₁₂−𝜔²₁𝑘=0,₀₁𝑘₁₁+𝑘₀₂𝑘₁₂−𝜔₀𝜔₁=0, (25) where the sign “±” in (22) and (25) takes “−” when 𝑚>0 and takes “+” when 𝑚<0. Due to the existence of the arbitrary functions, abundant exact solutions of (1) will be obtained as long as the function 𝑣(𝜉) is properly selected.

When we take 𝑣(𝜉)=𝜉arctan(𝜉), (26) a (2+1)-dimensional two-periodic solitoff solution of 2DsG equation can be obtained: Φ=4arctan√𝑛sn|𝑚|𝑘1+𝑛¹¹𝑥+𝑘₁₂𝑦+𝜔₁𝑡×arctan𝑘₁₁𝑥+𝑘₁₂𝑦+𝜔₁𝑡+𝑘₀₁𝑥+𝑘₀₂𝑦+𝜔₀𝑡. (27) 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 𝑣₁=𝜔₀/𝑘²₀₁+𝑘²₀₂≠1 and 𝑣₂=𝜔₁/𝑘²₁₁+𝑘²₁₂=1, respectively.

Figure 3 shows a two-periodic solitoff solution (27) with these parameters 𝑘₀₁=1,𝑘₀₂=3,𝑘₁₁=1.2,𝑘₁₂𝜔=1.6,₀=3,𝜔₁=2,𝑚=−1,𝑛=0.9 (28) at time 𝑡=0. The angle of the two-periodic solitoff in this figure is actually an obtuse angle although it seems to be orthogonal. It is because 𝑘₁⋅𝑘₂=𝑘₀₁𝑘₁₁+𝑘₀₂𝑘₁₂=𝜔₀𝜔₁≠0. 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.

Figure 3 

A two-periodic solitoff solution of 2DsG equation (27) with (28) at time 𝑡=0.

(a)

(b)

Figure 4 

The propagation of two-periodic solitoff solution (27) with (28) along the 𝑦-axis when 𝑥=0 at (a) 𝑡=−5 and (b) 𝑡=25.

A periodic soliton-periodic travelling wave interaction solution of 2DsG equation can be obtained: Φ=4arctan√𝑛sn|𝑚|𝑘1+𝑛⁰¹𝑥+𝑘₀₂𝑦+𝜔₀𝑡+7sech²𝑘₁₁𝑥+𝑘₁₂𝑦+𝜔₁𝑡+3, (29) by choosing 𝑣(𝜉)=7sech²(𝜉)+3. (30)

Figure 5(a) shows the periodic soliton-periodic travelling wave interaction solution (29) with these parameters 𝑘₀₁=1,𝑘₀₂=3,𝑘₁₁=3,𝑘₁₂𝜔=4,₀=3,𝜔₁=5,𝑚=−1,𝑛=0.9 (31) at time 𝑡=0. 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 Φ on the x-y plane.

(a)

(b)

Figure 5 

(a) A periodic soliton-periodic travelling wave interaction solution (29) with (31) at time 𝑡=0. (b) The density of Φ on the x-y plane.

Furthermore, if we take 𝑣(𝜉)=𝜉²5+1+4sin³(𝜉),𝑣(𝜉)=3cos³(𝜉)cn(𝜉)+1, (32) then a two-sawtooth-solitoff solution and a periodic solitoff-kink interaction solution of 2DsG equation can be written asΦ=4arctan√𝑛sn|𝑚|51+𝑛4sin³𝑘₁₁𝑥+𝑘₁₂𝑦+𝜔₁𝑡+𝑘₀₁𝑥+𝑘₀₂𝑦+𝜔₀𝑡+𝑘₁₁𝑥+𝑘₁₂𝑦+𝜔₁𝑡²,√+1Φ=4arctan𝑛sn|𝑚|𝑘1+𝑛⁰¹𝑥+𝑘₀₂𝑦+𝜔₀𝑡+1+3cos³𝑘₁₁𝑥+𝑘₁₂𝑦+𝜔₁𝑡×cn𝑘₁₁𝑥+𝑘₁₂𝑦+𝜔₁𝑡, (33)(34)respectively. Figure 6 shows a two-toothed-solitoff solution (33) with (31) in the limit case of the modulus 𝑛=1. 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.

(a)

(b)

Figure 6 

(a) A two-toothed-solitoff solution (33) with (31) at time 𝑡=1 except for the modulus 𝑛=1. (b) The density of Φ on the x-y plane.

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.

(a)

(b)

(c)

(d)

Figure 7 

(a) A periodic solitoff-kink interaction solution (34) with (28) at time 𝑡=0 except for the modulus 𝑛=1. (b)–(d) show the propagation of the periodic solitoff-kink interaction solution along the x-axis when 𝑦=−1.3 at 𝑡=−3, 𝑡=−0.5, and 𝑡=1.

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 (sn), more functions can be selected such as exp, cn, and cn/sn, 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.