The Multisoliton Solutions for the (2+1)-Dimensional Sawada-Kotera Equation

Abstract

Applying bilinear form and extended three-wavetype of ansätz approach on the (2+1)-dimensional Sawada-Kotera equation, we obtain new multisoliton solutions, including the double periodic-type three-wave solutions, the breather two-soliton solutions, the double breather soliton solutions, and the three-solitary solutions. These results show that the high-dimensional nonlinear evolution equation has rich dynamical behavior.

1. Introduction

As is well known that the exact solutions of nonlinear evolution equations play an important role in nonlinear science field, especially in nonlinear physical science since they can provide much physical information and more insight into the physical aspects of the problem and thus lead to further applications. The search for exact solutions of nonlinear partial differential equations has long been an interesting and hot topic in nonlinear mathematical physics. Consequently, many methods are available to look for exact solutions of nonlinear evolution equations, such as the inverse scattering method, the Lie group method, the mapping method, Exp-function method, and ansätz technique [1–4]. Very recently, Wang et al. [5] proposed a new technique called extended three-wave approach to seek multiwave solutions for integrable equations, and this method has been used to investigate several equations [6, 7]. In this paper, we consider the following Sawada-Kotera equation: 𝑢_𝑡=𝑢_{𝑥𝑥𝑥𝑥}+5𝑢𝑢_𝑥𝑥+53𝑢³+5𝑢_𝑥𝑦𝑥−5𝑢_𝑦𝑦𝑑𝑥+5𝑢𝑢_𝑦+5𝑢_𝑥𝑢_𝑦𝑑𝑥. (1) Equation (1) was derived by B. G. Konopelchenko and V. G. Dubrovsky, and was called the Sawada-Kotera (SK) equation; for example, see [8]. By means of the two-soliton method, the exact periodic soliton solutions, N-soliton solutions, and traveling wave solutions of the SK equation were found [8–10].

In this paper, we discuss further the (2+1)-dimensional SK equation, by using bilinear form and extended three-wave type of ansätz approach, respectively [5, 11–15], and some new multisoliton solutions are obtained.

2. The Multisoliton Solutions

We assume 𝑢=−2ln𝑓_𝑥𝑥, (2) where 𝑓=𝑓(𝑥,𝑦,𝑡) is an unknown real function. Substituting (2) into (1), we can reduce (1) into the following equation [8]: 𝐷⁶_𝑥+5𝐷_𝑦𝐷³_𝑥−5𝐷²_𝑦+𝐷_𝑥𝐷_𝑡𝑓⋅𝑓=0, (3) where the Hirota bilinear operator 𝐷 is defined by (𝑛,𝑚≥0)𝐷^𝑚_𝑥𝐷^𝑛_𝑡𝑓()()=𝜕𝑥,𝑡⋅𝑔𝑥,𝑡−𝜕𝜕𝑥𝜕𝑥^′𝑚𝜕−𝜕𝜕𝑡𝜕𝑡^′𝑛×𝑓(𝑥,𝑡)𝑔𝑥^′,𝑡^′_{𝑥′=𝑥,𝑡′=𝑡}. (4)

Now we suppose the solution of (3) as 𝑓=𝑒^−𝜉+𝛿₁cos𝜂+𝛿₂cosh𝛾+𝛿₃𝑒^𝜉, (5) where 𝜉=𝑎₁𝑥+𝑏₁𝑦+𝑐₁𝑡, 𝜂=𝑎₂𝑥+𝑏₂𝑦+𝑐₂𝑡, 𝛾=𝑎₃𝑥+𝑏₃𝑦+𝑐₃𝑡, and 𝑎_𝑖, 𝑏_𝑖, and 𝑐_𝑖 (𝑖=1,2,3) are some constants to be determined later. Substituting (5) into (3) and equating all the coefficients of different powers of 𝑒^𝜉, 𝑒^−𝜉, sin(𝜂), cos(𝜂), sinh(𝛾), cosh(𝛾), and the constant term to zero, we can obtain a set of algebraic equations for 𝑎_𝑖, 𝑏_𝑖, 𝑐_𝑖, and 𝛿_𝑗 (𝑖=1,2,3; 𝑗=1,2,3). Solving the system with the aid of Maple, we get the following results.

Case 1. If 𝑎₂=0, then 𝑏₁1=−4𝑎₁4𝑎²₁+3𝑎²₃,𝑏₂=32𝑖𝑎²₁𝑎₃,𝑏₃1=−4𝑎₃6𝑎²₁+𝑎²₃,𝛿₂𝛿=−¹𝑎²₁𝑎²₁−𝑎²₃,𝛿₃=𝛿₃,𝑐₁=9𝑎16¹5𝑎⁴₃+40𝑎²₁𝑎²₃+16𝑎⁴₁,𝑐₂=−454𝑖𝑎²₁𝑎₃2𝑎²₁+𝑎²₃,𝑐₃=9𝑎16³𝑎⁴₃+20𝑎²₁𝑎²₃+40𝑎⁴₁, (6) where 𝑎₁, 𝑎₃, 𝛿₁, and 𝛿₃ are free real constants. Substituting (6) into (5) and taking 𝛿₃>0, we have 𝑓₁=2𝛿₃cosh𝑎₁𝑥+𝐾₁𝑦+𝐿₁1𝑡+2ln𝛿₃−𝛿₁cosh𝑀₁𝑦+𝑁₁𝑡−𝛿₁𝑎²₁𝑎²₁−𝑎²₃×cosh𝑎₃𝑥−𝐻₁𝑦+𝐽₁𝑡, (7) where 𝐾₁=(1/4)𝑎₁(4𝑎²₁+3𝑎²₃), 𝐿₁=(9/16)𝑎₁(5𝑎⁴₃+40𝑎²₁𝑎²₃+16𝑎⁴₁), 𝑀₁=−(3/2)𝑎²₁𝑎₃, 𝑁₁=(45/4)𝑎²₁𝑎₃(2𝑎²₁+𝑎²₃), 𝐻₁=(1/4)𝑎₃(6𝑎²₁+𝑎²₃), and 𝐽₁=(9/16)𝑎₃(𝑎⁴₃+20𝑎²₁𝑎²₃+40𝑎⁴₁). Substituting (7) into (2) yields the three-soliton solution of SK equation as follows: 𝑢₁=−22𝛿₃𝑎²₁cosh𝜉₁+12ln𝛿₃−𝛿₁𝑎²₁𝑎²₃cosh𝜂₁𝑎²₁−𝑎²₃×2𝛿₃cosh𝜉₁+12ln𝛿₃−𝛿₁𝑎²₁cosh𝜂₁𝑎²₁−𝑎²₃−𝛿₁cosh𝛾₁⁻¹+22𝛿₃𝑎₁sinh𝜉₁+12ln𝛿₃−𝛿₁𝑎²₁𝑎₃sinh𝜂₁𝑎²₁−𝑎²₃×2𝛿₃cosh𝜉₁+12ln𝛿₃−𝛿₁𝑎²₁cosh𝜂₁𝑎²₁−𝑎²₃−𝛿₁cosh𝛾₁⁻¹², (8) where 𝜉₁=𝑎₁𝑥+𝐾₁𝑦+𝐿₁𝑡, 𝜂₁=𝑎₃𝑥−𝐻₁𝑦+𝐽₁𝑡, and 𝛾₁=𝑀₁𝑦+𝑁₁𝑡.
If taking 𝑎₃=𝑖𝐴₃ in (7), then we have 𝑓₂=2𝛿₃cosh𝑎₁𝑥+𝐾₂𝑦+𝐿₂1𝑡+2ln𝛿₃+𝛿₁cos𝑀₂𝑦+𝑁₂𝑡−𝛿₁𝑎²₁cos𝐴₃𝑥−𝐻₂𝑦+𝐽₂𝑡𝑎²₁+𝐴²₃, (9) where 𝛿₃>0, 𝐾₂=−(1/4)𝑎₁(4𝑎²₁−3𝐴²₃), 𝐿₂=(9/16)𝑎₁(5𝐴⁴₃−40𝑎²₁𝐴²₃+16𝑎⁴₁), 𝑀₂=(3/2)𝑎²₁𝐴₃, 𝑁₂=−(45/4)𝑎²₁𝐴₃(2𝑎²₁−𝐴²₃), 𝐻₂=𝐴₃𝑥−(1/4)𝐴₃(6𝑎²₁−𝐴²₃), and 𝐽₂=(9/16)𝐴₃(𝐴⁴₃−20𝑎²₁𝐴²₃+40𝑎⁴₁). Substituting (9) into (2) yields the double breather soliton solution of SK equation as follows: 𝑢₂=−22𝑎²₁𝛿₃cosh𝜉₂+12ln𝛿₃+𝛿₁𝑎²₁𝐴²₃cos𝜂₂𝑎²₁+𝐴²₃×2𝛿₃cosh𝜉₂+12ln𝛿₃+𝛿₁cos𝛾₂−𝛿₁𝑎²₁cos𝜂₂𝑎²₁+𝐴²₃⁻¹+22𝑎₁𝛿₃sinh𝜉₂+12ln𝛿₃+𝛿₁𝑎²₁𝐴₃sin𝜂₂𝑎²₁+𝐴²₃×2𝛿₃cosh𝜉₂+12ln𝛿₃+𝛿₁cos𝛾₂−𝛿₁𝑎²₁cos𝜂₂𝑎²₁+𝐴²₃⁻¹², (10) where 𝜉₂=𝑎₁𝑥+𝐾₂𝑦+𝐿₂𝑡, 𝜂₂=𝐴₃𝑥−𝐻₂𝑦+𝐽₂𝑡, and 𝛾₂=𝑀₂𝑦+𝑁₂𝑡.

Case 2. If 𝑎₂≠0, then 𝑏₁=−𝑎³₁,𝑏₂=𝑎³₂,𝑏₃=−𝑎³₃,𝛿₁=𝛿₁,𝛿₂=𝛿₂,𝛿₃=𝛿₃,𝑐₁=9𝑎⁵₁,𝑐₂=9𝑎⁵₂,𝑐₃=9𝑎⁵₃, (11) where 𝑎₁, 𝑎₂, 𝑎₃, 𝛿₁, 𝛿₂, and 𝛿₃ are free real constants. Substituting (11) into (5) and taking 𝛿₃>0, we have 𝑓₃=2𝛿₃cosh𝑎₁𝑥−𝑎³₁𝑦+9𝑎⁵₁1𝑡+2ln𝛿₃+𝛿₁cos𝑎₂𝑥+𝑎³₂𝑦+9𝑎⁵₂𝑡+𝛿₂cosh𝑎₃𝑥−𝑎³₃𝑦+9𝑎⁵₃𝑡. (12) Substituting (12) into (2) yields the breather two-soliton solution of SK equation as follows: 𝑢₃=−22𝛿₃𝑎²₁cosh𝜉₃+12ln𝛿₃−𝛿₁𝑎²₂cos𝜂₃+𝛿₂𝑎²₃cosh𝛾₃×2𝛿₃cosh𝜉₃+12ln𝛿₃+𝛿₁cos𝜂₃+𝛿₂cosh𝛾₃⁻¹+22𝛿₃𝑎₁sinh𝜉₃+12ln𝛿₃−𝛿₁𝑎₂sin𝜂₃+𝛿₂𝑎₃sinh𝛾₃×2𝛿₃cosh𝜉₃+12ln𝛿₃+𝛿₁cos𝜂₃+𝛿₂cosh𝛾₃⁻¹², (13) where 𝜉₃=𝑎₁𝑥−𝑎³₁𝑦+9𝑎⁵₁𝑡, 𝜂₃=𝑎₂𝑥+𝑎³₂𝑦+9𝑎⁵₂𝑡, and 𝛾₃=𝑎₃𝑥−𝑎³₃𝑦+9𝑎⁵₃𝑡.
The expression (𝑢₃) is the breather two-soliton solution of SK equation which is a periodic wave in 𝑥, 𝑦 and meanwhile is a two-soliton in 𝑥, 𝑦 (refer to Figure 1(b)).

(a)

(b)

(c)

(d)

Figure 1 

(a) The figure of 𝑢2 as 𝛿1=1, 𝛿3=1, and 𝑡=1. (b) The figure of 𝑢3 as 𝛿1=√2, 𝛿2=1, and 𝑡=0. (c) The figure of 𝑢4 as 𝛿1=√2, 𝛿2=√5, and 𝑡=0.005. (d) The figure of 𝑢5 as 𝛿1=1, 𝛿2=1, and 𝑡=0.

Case 3. If 𝑎₂=𝑏₁=0, then 𝑎₁=2𝑎₃,𝑏₂=√21𝑎³₃,𝑏₃3=−2𝑎³₃,𝑐₁=−1692𝑎⁵₃,𝑐₂=−20√21𝑎⁵₃,𝑐₃=−3494𝑎⁵₃,𝛿₃=5𝛿152²²−7𝛿228²¹, (14) where 𝑎₃, 𝛿₁, and 𝛿₂ are free real constants. Substituting (14) into (5) and taking 𝛿₃>0, we have 𝑓₄=25𝛿152²²−7𝛿228²¹×cosh−2𝑎₃𝑥+1692𝑎⁵₃1𝑡−25ln𝛿152²²−7𝛿228²¹+𝛿₁cos−√21𝑎³₃𝑦+20√21𝑎⁵₃𝑡+𝛿₂cosh−𝑎₃3𝑥+2𝑎₃³𝑦+3494𝑎⁵₃𝑡, (15) where (5/152)𝛿²₂−(7/228)𝛿²₁>0. Substituting (15) into (2) yields the breather two-soliton solution of SK equation as follows: 𝑢₄=−28𝐾₄𝑎²₃cosh𝜉₄−12ln𝐾₄+𝛿₂𝑎²₃cosh𝜂₄×2𝐾₄cosh𝜉₄−12ln𝐾₄+𝛿₁cos𝛾₄+𝛿₂cosh𝜂₄⁻¹+24𝐾₄𝑎₃sinh𝜉₄−12ln𝐾₄+𝛿₂𝑎₃sinh𝜂₄×2𝐾₄cosh𝜉₄−12ln𝐾₄+𝛿₁cos𝛾₄+𝛿₂cosh𝜂₄⁻¹², (16) where 𝐾₄=(5/152)𝛿²₂−(7/228)𝛿²₁, 𝜉₄=−2𝑎₃𝑥+(169/2)𝑎⁵₃𝑡, 𝜂₄=−𝑎₃𝑥+(3/2)𝑎³₃𝑦+(349/4)𝑎⁵₃𝑡, and 𝛾₄=−√21𝑎³₃𝑦+20√21𝑎⁵₃𝑡.
The expression (𝑢₄) is the breather two-soliton solution of SK equation which is a periodic wave in 𝑦-𝑡 and meanwhile is a two-soliton in 𝑥, 𝑦 and in 𝑥-𝑡, respectively (refer to Figure 1(c)).
Notice that 𝑢₃ and 𝑢₄ are also the breather two-soliton solutions, but their structure is different, because the two wave propagation directions are different in the 𝑢₃ and 𝑢₄, respectively (refer to Figures 1(b) and 1(c)).
If taking 𝑎₁=𝑖𝐴₁, 𝑎₃=𝑖𝐴₃ in (12), then we have 𝑓₅=2cos𝐴₁𝑥+𝐴³₁𝑦+9𝐴⁵₁𝑡+𝛿₁cos𝑎₂𝑥+𝑎³₂𝑦+9𝑎⁵₂𝑡+𝛿₂cos𝐴₃𝑥+𝐴³₃𝑦+9𝐴⁵₃𝑡, (17) when 𝛿₃=1. Substituting (17) into (2) gives the double-periodic three-wave solution of SK equation as follows: 𝑢₅=22𝐴²₁cos𝜉₅+𝛿₁𝑎²₂cos𝜂₅+𝛿₂𝐴²₃cos𝛾₅2cos𝜉₅+𝛿₁cos𝜂₅+𝛿₂cos𝛾₅+22𝐴₁sin𝜉₅+𝛿₁𝑎₂sin𝜂₅+𝛿₂𝐴₃sin𝛾₅2cos𝜉₅+𝛿₁cos𝜂₅+𝛿₂cos𝛾₅², (18) where 𝜉₅=𝐴₁𝑥+𝐴³₁𝑦+9𝐴⁵₁𝑡, 𝜂₅=𝑎₂𝑥+𝑎³₂𝑦+9𝑎⁵₂𝑡, and 𝛾₅=𝐴₃𝑥+𝐴³₃𝑦+9𝐴⁵₃𝑡.

3. Conclusion

By using bilinear form and extended three-wave type of ansätz approach, we discuss further the (2+1)-dimensional Sawada-Kotera equation and find some new multisoliton solutions. The result shows that the extended three-wave type of ansätz approach may provide us with a straightforward and effective mathematical tool for seeking multiwave solutions of high-dimensional nonlinear evolution equations.

Acknowledgments

This work was supported by Chinese Natural Science Foundation Grant nos. 11061028, 10971169. Sichuan Educational Science Foundation Grant no. 09zc008.