^{1, 2}

^{3}

^{1}

^{2}

^{3}

We use a generalized Cole-Hopf transformation to obtain a condition that allows us to find exact solutions for several forms of the general seventh-order KdV equation (KdV7). A remarkable fact is that this condition is satisfied by three well-known particular cases of the KdV7. We also show some solutions in these cases. In the particular case of the seventh-order Kaup-Kupershmidt KdV equation we obtain other solutions by some ansatzes different from the Cole-Hopf transformation.

During the last years scientists have seen a great interest in the investigation of nonlinear processes. The reason for this is that they appear in various branches of natural sciences and particularly in almost all branches of physics: fluid dynamics, plasma physics, field theory, nonlinear optics, and condensed matter physics. In this sense, the study of nonlinear partial differential equations NLPDEs and their solutions has great relevance today. Some analytical methods such as Hirota method [

The simplest classes of exact solutions to a given partial differential equation are those obtained from a

The general seventh-order Korteweg de Vries equation (KdV7) [

seventh-order Sawada-Kotera-Ito equation [

seventh-order Lax equation [

seventh-order Kaup-Kupershmidt equation [

Exact solutions for several forms of (

This paper is organized as follow: In Section

The main purpose of this section consists on establishing a polynomial equation involving the coefficients

From (

Substituting (

From equations (

Now, we substitute expression for

Now, suppose that

Direct calculations show that (

Suppose that

Sawada-Kotera-Ito equation (

Lax equation (

Kaup-Kupershmidt equation (

Figure

Graph of solution given by (

It may be verified that condition (

Now, let us assume that

Sawada-Kotera-Ito equation (

The second solution is Lax equation (

As we remarked in the previous section, the seventh-order Kaup-Kupershmidt equation (

an exp rational ansatz:

the tanh-coth ansatz [

From (

We change the

Graph of solution (

Graph of solution (

We may employ other methods to find exact solutions to the seventh-order Kaup-Kupershmidt equation. Thus, a solution to (

^{3}^{3} + 2268 # ^{2}

(*

0

In this work we have obtained two conditions associated to Cole-Hopf transformation (

The authors are grateful to their anonymous referees for their valuable help, time, objective analysis, and patience in the correction of this work. Their comments and observations helped them to simplify results of calculations, to clarify the key ideas, and to improve their manuscript. They also thank the editor for giving them the opportunity to publish this paper.