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Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 194329, 14 pages
http://dx.doi.org/10.1155/2010/194329
Research Article

Application of the Cole-Hopf Transformation for Finding Exact Solutions to Several Forms of the Seventh-Order KdV Equation

1Department of Mathematics, University of Caldas, Manizales Cll 65 # 26-10, A.A. 275, Colombia
2Department of Mathematics, National University of Colombia, Campus La Nubia, Manizales, Caldas, Colombia
3Department of Mathematics, National University of Colombia, Bogotá Cll 45, Cra 30, Colombia

Received 20 August 2009; Revised 5 November 2009; Accepted 28 January 2010

Academic Editor: Gradimir V. Milovanović

Copyright © 2010 Alvaro H. Salas and Cesar A. Gómez S.. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1. Introduction

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 [1] and scattering inverse method [2] have been used to solve some NLPDEs. However, the use of these analytical methods is not an easy task. Therefore, several computational methods have been implemented to obtain exact solutions for these models. It is clear that the knowledge of closed-form solutions of (NLPDEs) facilitates the testing of numerical solvers, helps physicists to better understand the mechanism that governs the physic models, provides knowledge of the physic problem, provides possible applications, and aids mathematicians in the stability analysis of solutions. The following are some of the most important computational methods used to obtain exact solutions to NLPDEs: the tanh method [3], the generalized tanh method [4, 5], the extended tanh method [6, 7], the improved tanh-coth method [8, 9], and the Exp-function method [10, 11]. Practically, there is no unified method that could be used to handle all types of nonlinear problems. All previous computational methods are based on the reduction of the original equation to equations in fewer dependent or independent variables. The main idea is to find such variables and, by passing to them, to obtain simpler equations. In particular, finding exact solutions of some partial differential equations in two independent variables may be reduced to the problem of finding solutions of an appropriate ordinary differential equation (or a system of ordinary differential equations). Naturally, the ordinary differential equation thus obtained does not give all solutions of the original partial differential equation, but provides a class of solutions with some specific properties.

The simplest classes of exact solutions to a given partial differential equation are those obtained from a traveling-wave transformation.

The general seventh-order Korteweg de Vries equation (KdV7) [12] reads

which has been introduced by Pomeau et al. [13] for discussing the structural stability of standard Korteweg de Vries equation (KdV) under a singular perturbation. Some well-known particular cases of (1.1) are the following:

(i)seventh-order Sawada-Kotera-Ito equation [12, 1416] (, , , , , , ): (ii)seventh-order Lax equation [12, 17] (, , , , , , ): (iii)seventh-order Kaup-Kupershmidt equation [12, 18] (, , , , , , ):

Exact solutions for several forms of (1.1) have been obtained by other authors using the Hirota method [19], the tanh-coth method [19], and He's variational iteration method [20]. However, the principal objective of this work consists on presenting a condition over the coefficients of (1.1) to obtain new exact traveling-wave solutions different from those in [19, 20] by using a Cole-Hopf transformation. We show that some of the previous models (1.2)–(1.4) satisfy this condition, and therefore in these cases we obtain new exact solutions for them.

This paper is organized as follow: In Section 2, we make use of a Cole-Hopf transformation to derive a condition over the coefficients of (1.1) for the existence of traveling-wave solutions. In Section 3 we present new exact solutions for several forms of (1.1) which satisfy the condition given in Section 2. In Section 4, new exact solutions to (1.4) are obtained using other anzatzes. Finally, some conclusions are given.

2. Using the Cole-Hopf Transformation

The main purpose of this section consists on establishing a polynomial equation involving the coefficients , , , , , , and of (1.1) that allows us to find traveling wave solutions (soliton and periodic solutions) using a special Cole-Hopf transformation. To this end, we seek solutions to (1.1) in the form , where

From (1.1) and (2.1) we obtain the following seventh-order nonlinear ordinary differential equation:

With the aim to find exact solutions to (2.2) we use the following Cole-Hopf transformation [21, 22]:

where and are arbitrary real constants.

Substituting (2.3) into (2.2), we obtain a polynomial equation in the variable . Equating the coefficients of the different powers of to zero, the following algebraic system is obtained:

From equations (2.4) and (2.5), we obtain

Now, we substitute expression for in (2.8) into (2.6) and (2.7) and solve them for and to obtain

Substitution of (2.10) and (2.11) into (2.9) gives

From this last equation we obtain, in particular,

or

These last two expressions are valid for

Suppose that Substituting this expression into (2.10) and (2.11) gives

Eliminating , and from (2.8), (2.16), and (2.17), we obtain the following polynomial equation in the variables , , , , , and :

where

We will call (2.18) the first discriminant equation. It is remarkable the fact that (1.2), (1.3), and (1.4) satisfy this equation. All seventh order equations that satisfy the first discriminant equation (2.18) may be solved exactly by means of the Cole-Hopf transformation (2.3).

Now, suppose that . Since then (2.14) holds. Reasoning in a similar way, from (2.14) we obtain the following second polynomial equation in terms of the coefficients of (1.1), which we shall call the second discriminant equation for (1.1) associated to Cole-Hopf transformation (2.3):

where

Direct calculations show that (1.2) and (1.3) satisfy not only the first discriminant equation (2.18), but also the second one. However, (1.4) does not satisfy the second discriminant equation (2.20). All seventh-order equations that satisfy (2.20) can be solved exactly by using the Cole-Hopf transformation (2.3).

3. Solutions to the KdV7

Suppose that Then (1.1) satisfies (2.18). In this case, we obtain the following solution to (1.1):

So that, from we can obtain explicit solutions to (1.2), (1.3), and (1.4). They are

(i)Sawada-Kotera-Ito equation (1.2): (ii)Lax equation (1.3): (iii)Kaup-Kupershmidt equation (1.4):

Figure 1 shows the graph of (3.6) for , , and

194329.fig.001
Figure 1: Graph of solution given by (3.6).

It may be verified that condition (2.15) holds when is given by either (3.2) or (3.5) since .

Now, let us assume that In this case, is given by (2.14). Then (1.1) satisfies (2.20). We obtain the following solution to (1.1):

So that, from we can obtain explicit solutions to (1.2) and (1.3). The first is

(i)Sawada-Kotera-Ito equation (1.2): where Restriction (1.2) guarantees condition (2.15). However, when we obtain following solution to (1.2): and when another solution is given by (ii)The second solution is Lax equation (1.3) for any real number .

4. Other Exact Solutions to the Seventh-Order Kaup-Kupershmidt Equation

As we remarked in the previous section, the seventh-order Kaup-Kupershmidt equation (1.4) does not satisfy the second discriminant equation (2.20). However, we can use other methods to obtain exact solutions different from those we already obtained, for instance,

(i)an exp rational ansatz: (ii)the tanh-coth ansatz [23]: where , , , , , , , , , and are constants.
4.1. Solutions by the Ansatz

From (1.4) and (4.1) we obtain a polynomial equation in the variable . Solving it, we get the solution

From (4.3) the following solutions are obtained:

Observe that solutions given by (3.6) and (4.4) coincide. Consider

4.2. Solutions by the Ansatz

We change the and functions to their exponential form and then we substitute (4.2) into (1.4). We obtain a polynomial equation in the variable . Equating the coefficients of the different powers of to zero results in an algebraic system in the variables , , , , , , and . Solving it with the aid of a computer, we obtain the following solutions of (1.4).

(i): (ii): Figure 2 shows the graph of for, and Figure 3 shows the graph of for, and(iii):
194329.fig.002
Figure 2: Graph of solution (4.7).
194329.fig.003
Figure 3: Graph of solution (4.8).
4.3. Other Exact Solutions

We may employ other methods to find exact solutions to the seventh-order Kaup-Kupershmidt equation. Thus, a solution to (1.4) of the form

is

In particular, taking we get

Following solution results from (4.12) by replacing with and with

The choice gives

Finally, by using the ansatze

we obtain

The correctness of the solutions given in this work may be checked with the aid of either Mathematica 7.0 or Maple 12. For example, to see that function defined by (4.12) is a solution to the seventh-order Kaup-Kupershmidt (1.4), we may use Mathematica 7.0 as follows:

In []:= kk7 = # + 2016 #3 # + 630 ( #)3 + 2268 # # # + 504 #2 # + 252 # # + 147 # # + 42 # # + # &;

(*This defines the differential operator associated with the seventh-order Kaup-Kupershmidt equation*)(*This defines the solution*)

In []:=Simplify [kk7 [TrigToExp [sol]]]

(*Here we apply the differential operator to the solution and then we make use of the Simplify command*)

Out []:= 0

5. Conclusions

In this work we have obtained two conditions associated to Cole-Hopf transformation (2.3) for the existence of exact solutions to the general KdV7. Two cases have been analyzed. The first one is given by (2.13) and the other by (2.14). The first case leads to first discriminant equation (2.18) which is satisfied by Sawada-Kotera-Ito equation (1.2), Lax equation (1.3), and Kaup-Kupershmidt equation (1.4). The second case gives the second discriminant equation (2.20), which is fulfilled only by (1.2) and (1.3). However, when both conditions (2.13) and (2.14) hold, we may obtain solutions different from those that result from the two cases we mentioned. We did not consider this last case. Other works that related the problem of finding exact solutions of nonlinear PDE's may be found in [24, 25].

Acknowledgments

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.

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