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

Authors: Zhenhui Xu; Hanlin Chen; Wei Chen
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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:
(1)ut=(uxxxx+5uuxx+53u3+5uxy)x-5∫(uyy)dx+5uuy+5ux∫(uy)dx.
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(2)u=-2(lnf)xx,
where f=f(x,y,t) is an unknown real function. Substituting (2) into (1), we can reduce (1) into the following equation [8]:
(3)(Dx6+5DyDx3-5Dy2+DxDt)f·f=0,
where the Hirota bilinear operator D is defined by (n,m≥0)(4)DxmDtnf(x,t)·g(x,t)=(∂∂x-∂∂x′)m(∂∂t-∂∂t′)n×[f(x,t)g(x′,t′)]|x′=x,t′=t.Now we suppose the solution of (3) as
(5)f=e-ξ+δ1cos(η)+δ2cosh(γ)+δ3eξ,
where ξ=a1x+b1y+c1t, η=a2x+b2y+c2t, γ=a3x+b3y+c3t, and ai, bi, and ci (i=1,2,3) are some constants to be determined later. Substituting (5) into (3) and equating all the coefficients of different powers of eξ, 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 ai, bi, ci, and δj (i=1,2,3; j=1,2,3). Solving the system with the aid of Maple, we get the following results.Case 1.
Ifa2=0, then
(6)b1=-14a1(4a12+3a32),b2=32ia12a3,b3=-14a3(6a12+a32),δ2=-δ1a12a12-a32,δ3=δ3,c1=916a1(5a34+40a12a32+16a14),c2=-454ia12a3(2a12+a32),c3=916a3(a34+20a12a32+40a14),
where a1, a3, δ1, and δ3 are free real constants. Substituting (6) into (5) and taking δ3>0, we have
(7)f1=2δ3cosh(a1x+K1y+L1t+12ln(δ3))-δ1cosh(M1y+N1t)-δ1a12a12-a32×cosh(a3x-H1y+J1t),
where K1=(1/4)a1(4a12+3a32), L1=(9/16)a1(5a34+40a12a32+16a14), M1=-(3/2)a12a3, N1=(45/4)a12a3(2a12+a32), H1=(1/4)a3(6a12+a32), and J1=(9/16)a3(a34+20a12a32+40a14). Substituting (7) into (2) yields the three-soliton solution of SK equation as follows:
(8)u1=-(δ1a12a32cosh(η1)a12-a322[δ1a12a32cosh(η1)a12-a322δ3a12cosh(ξ1+12ln(δ3))-δ1a12a32cosh(η1)a12-a32])×(-δ1a12cosh(η1)a12-a32-δ1cosh(γ1)2δ3cosh(ξ1+12ln(δ3))-δ1a12cosh(η1)a12-a32-δ1cosh(γ1))-1+[-δ1a12cosh(η1)a12-a32-δ1cosh(γ1))-1(2(δ1a12a3sinh(η1)(a12-a32)2δ3a1sinh(ξ1+12ln(δ3))-δ1a12a3sinh(η1)(a12-a32)))×(δ1a12cosh(η1)a12-a322δ3cosh(ξ1+12ln(δ3))-δ1a12cosh(η1)a12-a32-δ1cosh(γ1))-1]2,
where ξ1=a1x+K1y+L1t, η1=a3x-H1y+J1t, and γ1=M1y+N1t.
If takinga3=iA3 in (7), then we have
(9)f2=2δ3cosh(a1x+K2y+L2t+12ln(δ3))+δ1cos(M2y+N2t)-δ1a12cos(A3x-H2y+J2t)a12+A32,
where δ3>0, K2=-(1/4)a1(4a12-3A32), L2=(9/16)a1(5A34-40a12A32+16a14), M2=(3/2)a12A3, N2=-(45/4)a12A3(2a12-A32), H2=A3x-(1/4)A3(6a12-A32), and J2=(9/16)A3(A34-20a12A32+40a14). Substituting (9) into (2) yields the double breather soliton solution of SK equation as follows:
(10)u2=-(2[+δ1cos(γ2)-δ1a12cos(η2)a12+A32)-12a12δ3cosh(ξ2+12ln(δ3))+δ1a12A32cos(η2)a12+A32])×(δ1a12cos(η2)a12+A322δ3cosh(ξ2+12ln(δ3))+δ1cos(γ2)-δ1a12cos(η2)a12+A32)-1+2[(δ1a12A3sin(η2)a12+A322a1δ3sinh(ξ2+12ln(δ3))+δ1a12A3sin(η2)a12+A32)×(δ1a12cos(η2)a12+A322δ3cosh(ξ2+12ln(δ3))+δ1cos(γ2)-δ1a12cos(η2)a12+A32)-1]2,
where ξ2=a1x+K2y+L2t, η2=A3x-H2y+J2t, and γ2=M2y+N2t.Case 2.
Ifa2≠0, then
(11)b1=-a13,b2=a23,b3=-a33,δ1=δ1,δ2=δ2,δ3=δ3,c1=9a15,c2=9a25,c3=9a35,
where a1, a2, a3, δ1, δ2, and δ3 are free real constants. Substituting (11) into (5) and taking δ3>0, we have
(12)f3=2δ3cosh(a1x-a13y+9a15t+12ln(δ3))+δ1cos(a2x+a23y+9a25t)+δ2cosh(a3x-a33y+9a35t).
Substituting (12) into (2) yields the breather two-soliton solution of SK equation as follows:
(13)u3=-(2[2δ3a12cosh(ξ3+12ln(δ3))-δ1a22cos(η3)+δ2a32cosh(γ3)(ξ3+12ln(δ3))])×(2δ3cosh(ξ3+12ln(δ3))+δ1cos(η3)+δ2cosh(γ3)(ξ3+12ln(δ3)))-1+[+δ1cos(η3)+δ2cosh(γ3)(ξ3+12ln(δ3)))-1(2(2δ3a1sinh(ξ3+12ln(δ3))-δ1a2sin(η3)+δ2a3sinh(γ3)(ξ3+12ln(δ3))))×(2δ3cosh(ξ3+12ln(δ3))+δ1cos(η3)+δ2cosh(γ3)(ξ3+12ln(δ3)))-1]2,
where ξ3=a1x-a13y+9a15t, η3=a2x+a23y+9a25t, and γ3=a3x-a33y+9a35t.
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) The figure ofu2 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.
(a)
(b)
(c)
(d)Case 3.
Ifa2=b1=0, then
(14)a1=2a3,b2=21a33,b3=-32a33,c1=-1692a35,c2=-2021a35,c3=-3494a35,δ3=5152δ22-7228δ12,
where a3, δ1, and δ2 are free real constants. Substituting (14) into (5) and taking δ3>0, we have
(15)f4=25152δ22-7228δ12×cosh(-2a3x+1692a35t-12ln(5152δ22-7228δ12))+δ1cos(-21a33y+2021a35t)+δ2cosh(-a3x+32a33y+3494a35t),
where (5/152)δ22-(7/228)δ12>0. Substituting (15) into (2) yields the breather two-soliton solution of SK equation as follows:
(16)u4=-(2[8K4a32cosh(ξ4-12ln(K4))+δ2a32cosh(η4)(ξ4-12ln(K4))])×(2K4cosh(ξ4-12ln(K4))+δ1cos(γ4)+δ2cosh(η4)(ξ4-12ln(K4)))-1+2[+δ1cos(γ4)+δ2cosh(η4)(ξ4-12ln(K4)))-1(4K4a3sinh(ξ4-12ln(K4))+δ2a3sinh(η4)(ξ4-12ln(K4)))×(2K4cosh(ξ4-12ln(K4))+δ1cos(γ4)+δ2cosh(η4)(ξ4-12ln(K4)))-1]2,
where K4=(5/152)δ22-(7/228)δ12, ξ4=-2a3x+(169/2)a35t, η4=-a3x+(3/2)a33y+(349/4)a35t, and γ4=-21a33y+2021a35t.
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 thatu3 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 takinga1=iA1, a3=iA3 in (12), then we have
(17)f5=2cos(A1x+A13y+9A15t)+δ1cos(a2x+a23y+9a25t)+δ2cos(A3x+A33y+9A35t),
when δ3=1. Substituting (17) into (2) gives the double-periodic three-wave solution of SK equation as follows:
(18)u5=2[2A12cos(ξ5)+δ1a22cos(η5)+δ2A32cos(γ5)]2cos(ξ5)+δ1cos(η5)+δ2cos(γ5)+2[2A1sin(ξ5)+δ1a2sin(η5)+δ2A3sin(γ5)2cos(ξ5)+δ1cos(η5)+δ2cos(γ5)]2,
where ξ5=A1x+A13y+9A15t, η5=a2x+a23y+9a25t, and γ5=A3x+A33y+9A35t.


## 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.



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