# Darboux Transformation and Explicit Solutions for a Generalized Sawada-Kotera Equation

Authors: Guo-Liang He; Ting Su
Journal: ISRN Applied Mathematics
 (2013)

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## Abstract

A generalized Sawada-Kotera equation and its Lax pairs are proposed. With the help of the gauge transformation between spectral problems, a Darboux transformation for the generalized SK equation is constructed. As an application of the Darboux transformation, we give some explicit solutions of the generalized SK equation such as the rational solutions, soliton solutions, and periodic solutions.


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## Body

## 1. Introduction

The Sawada-Kotera (SK) equation(1)ut=-uxxxxx+15(uuxx-u3)x
was first proposed by Sawada and Kotera when they gave a method for finding N-soliton solutions of the KdV equation and the KdV-like equation [1]. In [2], Caudrey et al. showed that (1) was a member of a new hierarchy of KdV equations. The SK equation’s physical importance was illustrated by Aiyer et al. in [3]. Then, the equation has been investigated by many authors [4–8]. The aim of the present paper is using the Darboux transformation [9–12] to study a generalized SK equation:
(2)ut=-uxxxxx+15(uuxx-u3)x-15(vux)x-10vvx,vt=-vxxxxx+15(uvxx+vvx-3vu2)x+30(vux)xx.The present paper is organized as follows. In Section2, with the aid of the Lax pairs of the SK equation [13, 14] and extending them by adding one potential function, we propose a generalized SK equation and its Lax pairs. Based on the gauge transformation between spectral problems, we derive a Darboux transformation of the generalized SK equation. In Section 3, the Darboux transformation is applied to the generalized SK equation, by which explicit solutions (we have verified the correctness of the solutions by using the Mathematic 5.0.) of the generalized SK equation are derived, including rational solutions, soliton solutions, and periodic solutions.


## 2.  Darboux Transformation of the Generalized Sawada-Kotera Equation

In this section, we will derive a generalized SK equation and its Darboux transformation. To this end, we first introduce the Lax pairs:(3)ℒψ=λψ,ψt=ℬψ,
where operators ℒ and ℬ are defined as follows:
(4)ℒ=∂3-3u∂+v,ℬ=9∂5-45u∂3+15(v-3ux)∂2+15(3u2-2uxx+vx)∂+10(vxx-3uv).
Then the compatibility condition between the two equations of (3) yields the Lax equation, ℒt=[ℬ,ℒ], which is equivalent to the generalized SK equation:
(5)ut=-uxxxxx+15(uuxx-u3)x-15(vux)x-10vvx,vt=-vxxxxx+15(uvxx+vvx-3u2v)x+30(uxv)xx.
If we choose v=0 and v=-(3/2)ux, (5) can be, respectively, reduced to the SK equation:
(6)ut=-uxxxxx-45u2ux+15uuxxx+15uxuxx
and the Kaup-Kupershmidt equation:
(7)ut=-uxxxxx-45u2ux+15uuxxx+752uxuxx.Theorem 1.
Letf satisfy (3) with λ=λ0 and A=-(lnf)x. Then the following Darboux transformation gives the relation about the original solutions u, v of (5) and its new ones u-, v-:
(8)u-=u+Ax,v-=v-3ux+3AAx-3Axx.Proof.
Assume thatψ satisfies (3) and A=-(lnf)x. Let
(9)ψ-=ψx+Aψ.
Using the first expression of (3), a direct calculation gives the following equations:
(10)ψ-x=ψxx+Aψx+Axψ,ψ-xx=Aψxx+(3u+2Ax)ψx+(Axx-v+λ)ψ,ψ-xxx=3(u+Ax)ψxx+(3ux-v+3Axx+3uA+λ)ψx+(Axxx-vx-vA+λA)ψ.
Substituting (9) and (10) into the following equation:
(11)ℒ-ψ-=λψ-,
where
(12)ℒ-=∂3-3u-∂+v-
and comparing the coefficients of ψ,  ψx,  and ψxx, we obtain the following:
(13)u--u-Ax=0,3u-A-v--3ux+v-3Axx-3uA=0,(14)3u-Ax-v-A-Axxx+vx+vA=0.
Equation (13) implies the following:
(15)u-=u+Ax,v-=v-3ux+3AAx-3Axx.
Substituting (15) into (14) and integrating it once, we have the following:
(16)3uA+3AAx-A3-Axx+v=λ0,
where λ0 is a constant of integration. Through direct calculations, we arrive at the following:
(17)-fxxf=Ax-A2,-fxxxf=Axx-3AAx+A3,-f4xf=Axxx-4AAxx-3Ax2+6A2Ax-A4,-f5xf=Axxxx-5AAxxx-10AxAxx+15AAx2+10A2Axx-10A3Ax+A5.
Using (17) and A=-(lnf)x, a simple reduction shows that (16) gives rise to the following:
(18)ℒf=λ0f.
Similarly, we consider the following equation:(19)ψ-t=ℬ-ψ-,
where
(20)ℬ-=9∂5-45u-∂3+15(v--3u-x)∂2+15(3u-2-2u-xx+v-x)∂+10(v-xx-3u-v-).
Seeing (3), (8), and (9), a direct calculation shows that (19) gives the following:
(21)At=9Axxxxx-10vxxx-45uAxxx+30uvx+15vAxx-90Axx2+30uxv-90uxAxx+45u2Ax+90uAx2+90Ax3-75uxxAx-120AxAxxx+15vxAx+270AAxAxx+30A2Axxx-30AAxxxx-30uxxxA-90uA2Ax+90uAAxx+90uuxA-30vAAx-90A2Ax2+90uxAAx,
which together with (17) implies the following:
(22)ft=ℬf.
This means that both of the Lax pairs (3) and (11) and (19) have the same form; that is, they lead to the same equation (5). Therefore, original solutions u, v of the generalized SK equation (5) are mapped into its new ones u-, v- by the Darboux transformation (8).


## 3. Explicit Solutions of the Generalized Sawada-Kotera Equation

In this section, we will construct explicit solutions of the generalized SK equation (5) by using the Darboux transformation (8).(I) We choose a trivial solutionu=0, v=0 of (5). Then (3) with λ=λ0 is reduced to the following:
(23)ψxxx=λ0ψ,ψt=9ψxxxxx.
Let λ0=-k3(k≠0). We can see that (23) has a general solution:
(24)f=c1exp(Δ1)+c2exp(-12Δ1)cos(Δ2)+c3exp(-12Δ1)sin(Δ2),
where cj, (j=1,2,3) are constants and
(25)Δ1=-kx-9k5t,Δ2=32(kx-9k5t).
Using the Darboux transformation (8), we get an explicit solution of (5)(26)u-=3(c22+c32)k2exp(-3Δ1)-6c1k2exp(-(3/2)Δ1)[(c2+3c3)cos(Δ2)-(3c2-c3)sin(Δ2)]4[c1+c2exp(-(3/2)Δ1)cos(Δ2)+c3exp(-(3/2)Δ1)sin(Δ2)]2,v-=(-{9exp(-32Δ1)k3(4c12[(c2-3c3)cos(Δ2)+(3c2+c3)sin(Δ2)]+exp(-3Δ1)([(c2-3c3)cos(Δ2)+(3c2+c3)sin(Δ2)](c22+c32))+2exp(-32Δ1)c1((6+cos(2Δ2)-3sin(2Δ2))c22+2(sin(2Δ2)+3cos(2Δ2))c2c3+(6-cos(2Δ2)+3sin(2Δ2))c32))9exp(-32Δ1)k3})×(8[c1+c2exp(-32Δ1)cos(Δ2)+c3exp(-32Δ1)sin(Δ2)]3)-1.Particularly, when we choosec1=0, c2=c3=1, we can get a periodic solution of (5):
(27)u-=3k22[cos(Δ2)+sin(Δ2)]2,v-=-3k3+3k3(12+32cos(Δ2)-sin(Δ2)cos(Δ2)+sin(Δ2))3+3k32sin(2Δ2)-1[cos(Δ2)+sin(Δ2)]2.
Plots of the solutions are given in Figures 1 and 2.Figure 1
u
-.Figure 2
v
-.(II) We consider the trivial solutionu=0, v=1 of (5). Then (3) with λ=λ0 is reduced to the following:
(28)ψxxx=(λ0-1)ψ,ψt=9ψxxxxx+15ψxx.Case 1.
Whenλ0=1, it is easy to see that (28) has a general solution:
(29)f=αx2+βx+γ+30αt,
where α, β, and γ are constants. Using the Darboux transformation (8), we get a rational solution of the generalized SK equation (5):
(30)u-=2α2x2+2αβx+β2-2αγ-60α2t[αx2+βx+γ+30αt]2,v-=1+3(β2-4α(γ+30αt))(β+2αx)[αx2+βx+γ+30αt]3.Case 2.
Whenλ0=1-k3(k≠0), (28) has a general solution:
(31)f=c1exp(Δ1)+c2exp(-12Δ1)cos(Δ2)+c3exp(-12Δ1)sin(Δ2),
where cj, (j=1,2,3) are constants and
(32)Δ1=-kx-(9k5-15k2)t,Δ2=32(kx-9k5t+15k2t).
Using the Darboux transformation (8), we get an explicit solution of (5):(33)u-=3(c22+c32)k2exp(-3Δ1)-6c1k2exp(-(3/2)Δ1)[(c2+3c3)cos(Δ2)-(3c2-c3)sin(Δ2)]4[c1+c2exp(-(3/2)Δ1)cos(Δ2)+c3exp(-(3/2)Δ1)sin(Δ2)]2,v-=(1-{9exp(-32Δ1)k3(+(6-cos(2Δ2)+3sin(2Δ2))c32)9exp(-32Δ1)4c12[(c2-3c3)cos(Δ2)+(3c2+c3)sin(Δ2)]+exp(-3Δ1)([(c2-3c3)cos(Δ2)+(3c2+c3)sin(Δ2)](c22+c32))+2exp(-32Δ1)c1((6+cos(2Δ2)-3sin(2Δ2))c22+2(sin(2Δ2)+3cos(2Δ2))c2c3+(6-cos(2Δ2)+3sin(2Δ2))c32)9exp(-32Δ1))})×(8[c1+c2exp(-32Δ1)cos(Δ2)c3exp(-32Δ1)sin(Δ2)]3)-1.
(III) We choose another trivial solutionu=1, v=0 of (5). Then (3) with λ=λ0 is reduced to the following:
(34)ψxxx=3ψx+λ0ψ,ψt=9ψxxxxx-45ψxxx+45ψx.Case 1.
Forλ0=0, a direct calculation gives a general solution of (34):
(35)f=c1exp(Δ)+c2exp(-Δ),
where c1, c2 are constants and Δ=3x-93t. Using the Darboux transformation (8), we get a soliton solution of the generalized SK equation (5) (c1=c2=1):
(36)u-=3[tanh(Δ)]2-2,v-=93[tanh(Δ)]3-93tanh(Δ).
Plots of the solutions are given in Figures 3 and 4.Figure 3
u
-.Figure 4
v
-.Case 2.
Forλ0=k3-3k(k≠0), (34) has a general solution:
(37)f=c1exp(Δ1)+c2exp(Δ2)+c3exp(Δ3),
where cj, (j=1,2,3) are constants and
(38)Δ1=kx+(9k5-45k3+45k)t,Δ2=-k+12-3k22x-92(5k-5k3+k5+12-3k2-3k212-3k2+k412-3k2)t,Δ3=-k-12-3k22x+92(-5k+5k3-k5+12-3k2-3k212-3k2+k412-3k2)t.
Using the Darboux transformation (8), we get an explicit solution of (5):
(39)u-=1+322c2c3(k2-4)exp(Δ2+Δ3)-c1c2(2+k2-k12-3k2)exp(Δ1+Δ2)-c1c3(2+k2+k12-3k2)exp(Δ1+Δ3)[c1exp(Δ1)+c2exp(Δ2)+c3exp(Δ3)]2,v-=3k3-9k+3Δ[c1exp(Δ1)+c2exp(Δ2)+c3exp(Δ3)]3,
where
(40)Δ=k(3-k2)[c13exp(3Δ1)+c23exp(3Δ2)+c33exp(3Δ3)-12c1c2c3exp(Δ1+Δ2+Δ3)c13exp(3Δ1)]+32c22c3exp(2Δ2+Δ3)×[2k-k3-412-3k2+k212-3k2]+32c2c32exp(Δ2+2Δ3)×[2k-k3+412-3k2-k212-3k2]-32c12c2exp(2Δ1+Δ2)(1+k2)(k-12-3k2)-32c12c3exp(2Δ1+Δ3)(1+k2)(k+12-3k2)+32c1c32exp(Δ1+2Δ3)k(8-k2+k12-3k2)+32c1c22exp(Δ1+2Δ2)k(8-k2-k12-3k2).



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