# Exact Solutions to KdV6 Equation by Using a New Approach of the Projective Riccati Equation Method

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

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## Abstract

We study a new integrable KdV6 equation from the point of view of its exact solutions by using an improved computational method.  A new approach to the projective Riccati equations method is implemented and used to construct  traveling wave solutions for a new integrable system, which is equivalent to KdV6 equation.  Periodic and soliton solutions are formally derived. Finally, some conclusions are given.


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## Body

## 1. Introduction

The sixth-order nonlinear wave equation(1.1)(∂x3+8ux∂x+4uxx)(ut+uxxx+6ux2)=0
has been recently derived by Karasu-Kalkanl1 et al. [1] as a new integrable particular case of the general sixth-order wave equation(1.2)uxxxxxx+αuxuxxxx+βuxxuxxx+γux2uxx+δutt+ρuxxxt+ωuxuxt+σutuxx=0,
where, α, β, γ, δ, ρ, ω, σ are arbitrary parameters, and u=u(x,t), is a differentiable function. By means of the change of variable(1.3)v=ux,w̃=ut+uxxx+6ux2,
equation (1.1) converts to the Korteweg-de Vries equation with a source satisfying a third-order ordinary differential equation (KdV6)(1.4)vt+vxxx+12vvx-w̃x=0,w̃xxx+8vw̃x+4w̃vx=0,
which is regarded as a nonholonomic deformation of the KdV equation [2]. Setting(1.5)v(x,t)=12u(x,-t),w̃(x,t)=12w(x,t),
the system (1.4) reduces to [2, 3] (1.6)ut-6uux-uxxx+wx=0,wxxx+4uwx+2uxw=0.
A first study on the integrability of (1.6) has been done by Kupershmidt [2]. However, only at the end of the last year, Yao and Zeng [4] have derived the integrability of (1.6). More exactly, they showed that (1.6) is equivalent to the Rosochatius deformations of the KdV equation with self-consistent sources (RD-KdVESCS). This is a remarkable fact because the soliton equations with self-consistent sources (SESCS) have important physical applications. For instance, the KdV equation with self-consistent sources (KdVESCS) describes the interaction of long and short capillary-gravity waves [5]. On the other hand, when w=0 the system (1.6) reduces to potential KdV equation, so that solutions of the potential KdV equation are solutions to (1.1). Furthermore, solving (1.6) we can obtain new solutions to (1.1). In the soliton theory, several computational methods have been implemented to handle nonlinear evolution equations. Among them are the tanh method [6], generalized tanh method [7, 8], the extended tanh method [9–11], the improved tanh-coth method [12, 13], the Exp-function method [14–16], the projective Riccati equations method [17], the generalized projective Riccati equations method [18–23], the extended hyperbolic function method [24], variational iteration method [25–27], He's polynomials [28], homotopy perturbation method [29–31], and many other methods [32–35], which have been used in a satisfactory way to obtain exact solutions to NLPDEs. Exact solutions to system (1.6) and (1.1) have been obtained using several methods [3, 4, 36–38]. In this paper, we obtain exact solutions to system (1.6). However, our idea is based on a new version of the projective Riccati method which can be considered as a generalized method, from which all other methods can be derived. This paper is organized as follows. In Section 2 we briefly review the new improved projective Riccati equations method. In Section 3 we give the mathematical framework to search exact for solutions to the system (1.6). In Section 4, we mention a new sixth-order KdV system from which novel solutions to (1.6) can be derived. Finally, some conclusions are given.


## 2. The Method

In the search of the traveling wave solutions to nonlinear partial differential equation of the form(2.1)P(u,ux,ut,uxx,uxt,utt,…)=0,
the first step consists in use the wave transformation(2.2)u(x,t)=v(ξ),ξ=x+λt,
where λ is a constant. With (2.2), equation (2.1) converts to an ordinary differential equation (ODE) for the function v(ξ)(2.3)P(v,v′,v′′,…)=0.
To find solutions to (2.3), we suppose that v(ξ) can be expressed as(2.4)v(ξ)=H(f(ξ),g(ξ)),
where H(f(ξ),g(ξ)) is a rational function in the new variables f(ξ), g(ξ) which are solutions to the system (2.5)f′(ξ)=ρf(ξ)g(ξ),g2(ξ)=R(f(ξ)),
being ρ≠0 an arbitrary constant to be determinate and R(f(ξ)) a rational function in the variable f(ξ). Taking(2.6)f(ξ)=ϕN(ξ),
where ϕ(ξ)≠0, and N≠0, then (2.5) reduces to (2.7)ϕ′(ξ)=ρNϕ(ξ)g(ξ),g2(ξ)=R(ϕN(ξ)).
From (2.7) we obtain(2.8)(ϕ′(ξ))2=ρ2N2ϕ2(ξ)R(ϕN).LetN=-1 and R(f(ξ))=α+βf(ξ)+γf(ξ)2, with α≠0. In this case, (2.8) reduces to(2.9)(ϕ′(ξ))2=ρ2ϕ(ξ)2R(ϕ(ξ)-1)=ρ2(αϕ2(ξ)+βϕ(ξ)+γ),
and (2.5) are transformed into (2.10)f′(ξ)=ρf(ξ)g(ξ),g2(ξ)=α+βf(ξ)+γf(ξ)2.
The following are solutions to (2.9): (2.11)ϕ1(ξ)=14α(-2β+(1-Δ)sinh(ραξ)+(1+Δ)cosh(ραξ)),ϕ2(ξ)=14α(-2β-(1-Δ)sinh(ραξ)+(1+Δ)cosh(ραξ)).
Therefore, solutions to (2.10) are given by (2.12)f(ξ)=-4α2β±(1-Δ)sinh(ραξ)-(1+Δ)cosh(ραξ),g(ξ)=α((1+Δ)sinh(ραξ)∓(1-Δ)cosh(ραξ))2β±(1-Δ)sinh(ραξ)-(1+Δ)cosh(ραξ).
In all cases Δ=β2-4αγ.


## 3. Exact Solutions to the Integrable KdV6 System

Using the traveling wave transformation(3.1)u(x,t)=v(ξ),w(x,t)=w(ξ),ξ=x+λt+ξ0,
the system (1.6) reduces to
(3.2)(λv(ξ)-3v2(ξ)-v′′(ξ)+w(ξ))′=0,(3.3)w′′′(ξ)+4v(ξ)w'(ξ)+2v'(ξ)w(ξ)=0.
Integrating (3.2) with respect to ξ and setting the constant of integration to zero we obtain(3.4)λv(ξ)-3v2(ξ)-v′′(ξ)+w(ξ)=0,w′′′(ξ)+4v(ξ)w'(ξ)+2v'(ξ)w(ξ)=0.
Using the idea of the projective Riccati equations method [19–22], we seek solutions to (3.4) as follows:(3.5)v(ξ)=H1(f(ξ),g(ξ))=∑0Maifi(ξ)+∑M+12Maig(ξ)fi-(M+1)(ξ),w(ξ)=H2(f(ξ),g(ξ))=∑0Nbifi(ξ)+∑N+12Nbig(ξ)fi-(N+1)(ξ),
where f(ξ) and g(ξ) satisfy the system given by (2.10) (with ρ=1). Substituting (3.5) into (3.4), after balancing we have that(3.6)M=2,
and Nis an arbitrary positive constant. By simplicity we take N=M. Therefore, (3.5) reduce to (3.7)v(ξ)=H1(f(ξ),g(ξ))=∑02aifi(ξ)+∑34aig(ξ)fi-(3)(ξ),w(ξ)=H2(f(ξ),g(ξ))=∑02bifi(ξ)+∑34big(ξ)fi-(3)(ξ).
Substituting this last two equations into (3.4), using (2.10) we obtain an algebraic system in the unknowns a0, a1, a2, a3, a4, b0, b1, b2, b3, b4, λ, α, β, and γ. Solving it and using (3.7), (2.12), and (3.1) we have the following set of new nontrivial solutions to KdV6 system (1.6). In all cases, a1=a3=b1=b3=β=0(3.8)λ=2b4∓α2a42+9b42a4,b0=16(-αb4a4-6b42a42±2b4α2a42+9b42a42),b2=-a4b4,a0=-αa4+3b4∓α2a42+9b426a4,a2=-a42,γ=a42.
A combined formal soliton solution is:(3.9)u1(x,t)=-αa4+3b4∓α2a42+9b426a4+(-a42)(-4α±(1+4αa42)sinh[αξ]-(1-4αa42)cosh[αξ])2+a4(-4α±(1+4αa42)sinh[αξ]-(1-4αa42)cosh[αξ])×(α((1-4αa42)sinh[αξ]∓(1+4αa42)cosh[αξ])±(1+4αa42)sinh[αξ]-(1-4αa42)cosh[αξ]),w1(x,t)=16(-αb4a4-6b42a42±2b4α2a42+9b42a42)+(-a4b4)(-4α±(1+4αa42)sinh[αξ]-(1-4αa42)cosh[αξ])2×(α((1-4αa42)sinh[αξ]∓(1+4αa42)cosh[αξ])±(1+4αa42)sinh[αξ]-(1-4αa42)cosh[αξ])'
where a4,b4,α are arbitrary constants, and ξ=x+λt+ξ0.Furthermore,(3.10)λ=8α2+20αa0+15a022α+3a0,b0=-2(4α2a0+7αa02+3a03)2α+3a0,b4=0,a4=0,b2=-2(4αγa0+3γa02)2α+3a0,a2=-2γ.
A soliton solution is given by(3.11)u2(x,t)=a0+(-2γ)(-4α±(1+4αγ)sinh[αξ]-(1-4αγ)cosh[αξ])2,w2(x,t)=-2(4α2a0+7αa02+3a03)2α+3a0+(-2(4αγa0+3γa02)2α+3a0)×(-4α±(1+4αγ)sinh[αξ]-(1-4αγ)cosh[αξ])2,
where a0,α,γ are arbitrary constants and ξ=x+λt+ξ0.


### 3.1. A New System

A direct calculation shows that (1.1) reduces to(3.12)uxxxxxx+20uxuxxxx+40uxxuxxx+120ux2uxx+uxxxt+4uxxut+8uxuxt=0.
On the other hand, it is easy to see that (3.12) can be written as(3.13)(∂x2+4uxx∂x-1+8ux)(uxt+uxxxx+12uxuxx)=0.
Using the analogy between KdV equation and MKdV equation and motivated by the structure of (3.13), the authors in [38] have introduced the so-called MKdV6 equation(3.14)(∂x3+8vx2∂x+8vxx∂x-1vx∂x)(vt+vxxx+4vx3)=0,
and they showed that(3.15)(∂x3+8ux∂x+4uxx)(ut+uxxx+6ux2)=(2vx+22i∂x),(∂x3+8vx2∂x+8vxx∂x-1vx∂x)(vt+vxxx+4vx3)=0,
where vx2+2/2ivxx is the Miura transformation between  KdV6 equation (1.1) and  MKdV6 equation (3.14). Therefore, solving (3.14), according to (3.15), solutions to (1.1) are obtained. Setting wx=vx2, then the new MKdV6 equation is equivalent to new system(3.16)vxxxxxx+20vx2vxxxx+80vxvxxvxxx+20vxx3+120vx4vxx+vxxxt+8vx2vxt+4vxxwt=0,wxx-2vxvxx=0.
In equivalent form, with s=vx, w=vt+vxxx+4vx3, from (3.14) the following system is derived:(3.17)st+sxxx+12s2sx-wx=0,wxxx+8s2wx+8sxz=0,zx-swx=0.
We believe that traveling wave solutions to these systems can be obtained using the method used here. By reasons of space, we omit them.


## 3.1. A New System

A direct calculation shows that (1.1) reduces to(3.12)uxxxxxx+20uxuxxxx+40uxxuxxx+120ux2uxx+uxxxt+4uxxut+8uxuxt=0.
On the other hand, it is easy to see that (3.12) can be written as(3.13)(∂x2+4uxx∂x-1+8ux)(uxt+uxxxx+12uxuxx)=0.
Using the analogy between KdV equation and MKdV equation and motivated by the structure of (3.13), the authors in [38] have introduced the so-called MKdV6 equation(3.14)(∂x3+8vx2∂x+8vxx∂x-1vx∂x)(vt+vxxx+4vx3)=0,
and they showed that(3.15)(∂x3+8ux∂x+4uxx)(ut+uxxx+6ux2)=(2vx+22i∂x),(∂x3+8vx2∂x+8vxx∂x-1vx∂x)(vt+vxxx+4vx3)=0,
where vx2+2/2ivxx is the Miura transformation between  KdV6 equation (1.1) and  MKdV6 equation (3.14). Therefore, solving (3.14), according to (3.15), solutions to (1.1) are obtained. Setting wx=vx2, then the new MKdV6 equation is equivalent to new system(3.16)vxxxxxx+20vx2vxxxx+80vxvxxvxxx+20vxx3+120vx4vxx+vxxxt+8vx2vxt+4vxxwt=0,wxx-2vxvxx=0.
In equivalent form, with s=vx, w=vt+vxxx+4vx3, from (3.14) the following system is derived:(3.17)st+sxxx+12s2sx-wx=0,wxxx+8s2wx+8sxz=0,zx-swx=0.
We believe that traveling wave solutions to these systems can be obtained using the method used here. By reasons of space, we omit them.


## 4. Conclusions

In this paper we have derived two new soliton solutions to KdV6 system (1.2) by using a new approach of the improved projective Riccati equations method. The results show that the method is reliable and can be used to handle other NLPDE's. Other methods such as tanh, tanh-coth, and exp-function methods can be derived from the new version of the projective Riccati equation method. Moreover, new methods can be obtained using the exposed ideas in the present paper. Other methods related to the problem of solving nonlinear PDEs exactly may be found in [39, 40].



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*Source: 797084-2010-08-31.xml*