# Computing Exact Solutions to a Generalized Lax-Sawada-Kotera-Ito Seventh-Order KdV Equation

Authors: Alvaro H. Salas; Cesar A. Gómez S; Bernardo Acevedo Frias
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

The Cole-Hopf transform is used to construct exact solutions to a generalization of both the seventh-order Lax KdV equation (Lax KdV7) and the seventh-order Sawada-Kotera-Ito KdV equation (Sawada-Kotera-Ito KdV7).


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## Body

## 1. Introduction

Many direct and computational methods have been used to handle nonlinear partial differential equations (NLPDE’s). Some methods used in a satisfactory way to obtain exact solutions to NLPDE’s are inverse scattering method [1], Hirota bilinear method [2, 3], Backlund transformations [4], Painlevé analysis [5], Lie groups [6], the tanh method [7], the generalized tanh method [8, 9], the extended tanh method [10–12], the improved tanh-coth method [13, 14], the Exp-function method [15–17], the projective Riccati equation method [18], the generalized projective Riccati equations method [19–24], the extended hyperbolic function method [25], variational iteration method [26, 27], He’s polynomials [28], homotopy perturbation method [29], and many other methods [30]. However, there is not a unified method that could be used to handle all NLPDE’s; in this sense, the implementation of new methods or variants of the some well-known methods is relevant. The principal objective of this paper consists in obtaining exact traveling wave solutions which include periodic and soliton solutions to a particular case of the general seventh-order KdV(KdV7), which is a generalization of the seventh-order Sawada-Kotera-Ito (SKI-KdV7) equation, by using a variant of the exp-function method. The general seventh-order KdV (KdV7) equation [31] reads (1.1)ut+au3ux+bux3+cuuxuxx+du2uxxx+eu2xu3x+fuxu4x+guu5x+u7x=0.
The (KdV7) was introduced initially by Pomeau et al. [32] for discussing the structural stability of KdV equation under a singular perturbation. Some particular cases of  (1.1) are (i)
seventh-order Lax equation [1, 6] (a=140,b=70,c=280,d=70,e=70,f=42,g=14):
(1.2)ut+140u3ux+70ux3+280uuxuxx+70u2uxxx+70u2xu3x+42uxu4x+14uu5x+u7x=0;(ii)
seventh-order Sawada-Kotera-Ito equation [1, 8–10] (a=252,b=63,c=378,d=126,e=63,f=42,g=21):
(1.3)ut+252u3ux+63ux3+378uuxuxx+126u2uxxx+63u2xu3x+42uxu4x+21uu5x+u7x=0;(iii)
seventh-order Kaup-Kupershmidt equation [1, 7] (a=2016,b=630,c=2268,d=504,e=252,f=147,g=42):
(1.4)ut+2016u3ux+630ux3+2268uuxuxx+504u2uxxx+252u2xu3x+147uxu4x+42uu5x+u7x=0.


## 2. Generalization of the Lax KdV7 and the Sawada-Kotera-Ito KdV7

Observe that (1.2) and (1.3) satisfy the relation(2.1)a=d63(e+f+g).
For this reason we will study equation(2.2)ut+d63(e+f+g)u3ux+bux3+cuuxuxx+du2uxxx+eu2xu3x+fuxu4x+guu5x+u7x=0.
We seek solutions to (2.2) in the Cole-Hopf form (2.3)u(t,x)=A∂xtanh(ξ),
where A is some constant to be determined later and (2.4)ξ=ξ(t,x)=μ(x+λt+δ),μ,δ,λ=const.
Substituting (2.3) into (2.2), we obtain a polynomial equation in the variable ζ=exp(ξ). Equating the coefficients of the different powers of ζ to zero, we obtain following algebraic system:(2.5)λ+64μ6=0,64μ5(A(e+f+g)-247μ)+5λ=0,64μ4(A2(b+c+d)-3Aμ(5e+9f+19g)+4293μ2)+9λ=0,64μ3(A3d(e+f+g)-63A2μ(3b+5c+11d)+126Aμ2(28e+46f+151g)-983997μ3)+315λ=0.
Eliminating A,λ, and μ from system (2.5) gives(2.6)b=d+1126(e+f+g)(e-5f+10g),c=521g(e+f+g)-2d.
It is easy to verify that (1.2) and (1.3) are particular cases of general KdV7 equation (1.1) subject to (2.1) and (2.6). This motivates us to define the generalized Lax-Sawada-Kotera-Ito seventh-order equation  (LSKI KdV7) as follows:(2.7)ut+163d(e+f+g)u3ux+(d+1126(e+f+g)(e-5f+10g))ux3+(521g(e+f+g)-2d)uuxuxx+du2uxxx+eu2xu3x+fuxu4x+guu5x+u7x=0.


## 3. Solutions to Generalized LSKI KdV7

In order to look for solutions to (2.7), we will use the exp ansatz (3.1)u(ξ)=p+q1+rexp(-ξ)+sexp(ξ),
where p, q, r, and s are some constants. Substituting (3.1) into (2.7) gives an algebraic system. Solving it, we obtain
(3.2)λ=-163d(e+f+g)p3-μ2(dp2+gpμ2+μ4),q=126μ2e+f+g,s=14r,r=r,μ=μ.
From (2.4), (3.1), and (3.2), we obtain following solution to (2.7) subject:(3.3)u(x,t)=p+126μ2(e+f+g)(1+rexp(ξ)+(1/4r)exp(-ξ)),ξ=μ(x+λt+δ),λ=-163d(e+f+g)p3-μ2(dp2+gpμ2+μ4).
In particular, if r=1/2, equation (3.3) gives(3.4)u(x,t)=p+63μ2e+f+gsech2(μ2(x+λt+δ)),λ=-163d(e+f+g)p3-(dp2+gpμ2+μ4)μ2.
Replacing μ with μ-1 gives the following periodic solutions: (3.5)u(x,t)=p-63μ2e+f+gsec2(μ2(x+λt+δ)),λ=-163d(e+f+g)p3+(dp2-gpμ2+μ4)μ2.
On the other hand, if r=-1/2, equation (3.3) gives(3.6)u(x,t)=p-63μ2e+f+gcsch2(μ2(x+λt+δ)),λ=-163d(e+f+g)p3-(dp2+gpμ2+μ4)μ2.
Replacing μ with μ-1 gives the following periodic solutions:(3.7)u(x,t)=p-63μ2e+f+gcsc2(μ2(x+λt+δ)),λ=-163d(e+f+g)p3+(dp2-gpμ2+μ4)μ2.


## 4. Solutions to Sawada-Kotera-Ito KdV7 Equation

From (3.3)–(3.7) with d=126,e=63,f=42, and g=21, we obtain the following analytic solutions to equation (1.3): (4.1)u(x,t)=p+4rμ2exp(μ(x+λt+δ))(1+2rexp(μ(x+λt+δ)))2,λ=-252p3-126p2μ2-21pμ4-μ6,u(x,t)=p+12μ2sech2(12μ(x+λt+δ)),λ=-252p3-126p2μ2-21pμ4-μ6,u(x,t)=p-12μ2sec2(12μ(x+λt+δ)),λ=-252p3+126p2μ2-21pμ4+μ6,u(x,t)=p-12μ2csch2(12μ(x+λt+δ)),λ=-252p3-126p2μ2-21pμ4-μ6.


## 5. Conclusions

We exhibited an equation that generalizes both seventh-order Lax equation and seventh-order Sawada-Kotera-Ito equation. At the same time, we obtained exact solutions to these equations with the aid of a Cole-Hopf ansatz. These same ideas are suitable for the seventh-order Kaup-Kupershmidt equation. We think that some of the solutions in this work are new in the open literature. We may apply other methods to find exact solutions to a variety of nonlinear PDE’s. See [3, 12–52].



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*Source: 524567-2010-08-31.xml*