Research Article | Open Access

Alvaro H. Salas, Cesar A. Gómez S., "Application of the Cole-Hopf Transformation for Finding Exact Solutions to Several Forms of the Seventh-Order KdV Equation", *Mathematical Problems in Engineering*, vol. 2010, Article ID 194329, 14 pages, 2010. https://doi.org/10.1155/2010/194329

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

**Academic Editor:**Gradimir V. Milovanović

#### 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, 14–16] (, , , , , , ): (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

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):##### 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:

*∂*# + 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.

#### References

- R. Hirota, “Exact solutions of the KdV equation for multiple collisions of solitons,”
*Physical Review Letters*, vol. 23, p. 695, 1971. View at: Google Scholar - M. J. Ablowitz and P. A. Clarkson,
*Solitons, Nonlinear Evolution Equations and Inverse Scattering*, vol. 149 of*London Mathematical Society Lecture Note Series*, Cambridge University Press, Cambridge, UK, 1991. View at: Zentralblatt MATH | MathSciNet - E. Fan and Y. C. Hon, “Generalized tanh method extended to special types of nonlinear equations,”
*Zeitschrift für Naturforschung*, vol. 57, no. 8, pp. 692–700, 2002. View at: Google Scholar - C. A. Gómez, “Exact solutions for a new fifth-order integrable system,”
*Revista Colombiana de Matemáticas*, vol. 40, no. 2, pp. 119–125, 2006. View at: Google Scholar | MathSciNet - A. H. Salas and C. A. Gómez, “Exact solutions for a reaction diffusion equation by using the generalized tanh method,”
*Scientia et Technica*, vol. 13, pp. 409–410, 2007. View at: Google Scholar - A. M. Wazwaz, “The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations,”
*Applied Mathematics and Computation*, vol. 184, no. 2, pp. 1002–1014, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. A. Gómez, “Special forms of the fifth-order KdV equation with new periodic and soliton solutions,”
*Applied Mathematics and Computation*, vol. 189, no. 2, pp. 1066–1077, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. A. Gómez and A. H. Salas, “The generalized tanh-coth method to special types of the fifth-order KdV equation,”
*Applied Mathematics and Computation*, vol. 203, no. 2, pp. 873–880, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. H. Salas and C. A. Gómez, “Computing exact solutions for some fifth KdV equations with forcing term,”
*Applied Mathematics and Computation*, vol. 204, no. 1, pp. 257–260, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J.-H. He and X.-H. Wu, “Exp-function method for nonlinear wave equations,”
*Chaos, Solitons and Fractals*, vol. 30, no. 3, pp. 700–708, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Zhang, “Exp-function method exactly solving the KdV equation with forcing term,”
*Applied Mathematics and Computation*, vol. 197, no. 1, pp. 128–134, 2008. View at: Publisher Site | Google Scholar | MathSciNet - Ü. Göktaş and W. Hereman, “Symbolic computation of conserved densities for systems of nonlinear evolution equations,”
*Journal of Symbolic Computation*, vol. 24, pp. 591–622, 1997. View at: Google Scholar - Y. Pomeau, A. Ramani, and B. Grammaticos, “Structural stability of the Korteweg-de Vries solitons under a singular perturbation,”
*Physica D*, vol. 31, no. 1, pp. 127–134, 1988. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Ito, “An extension of nonlinear evolution equations of the K-dV (mK-dV) type to higher orders,”
*Journal of the Physical Society of Japan*, vol. 49, no. 2, pp. 771–778, 1980. View at: Publisher Site | Google Scholar | MathSciNet - P. J. Caudrey, R. K. Dodd, and J. D. Gibbon, “A new hierarchy of Korteweg-de Vries equations,”
*Proceedings of the Royal Society of London A*, vol. 351, no. 1666, pp. 407–422, 1976. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. K. Sawada and T. Kotera, “A method for finding $N$-soliton solutions of the KdV equation and KdV-like equation,”
*Progress of Theoretical Physics*, vol. 51, pp. 1355–1367, 1974. View at: Publisher Site | Google Scholar | MathSciNet - P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves,”
*Communications on Pure and Applied Mathematics*, vol. 21, pp. 467–490, 1968. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D. J. Kaup, “On the inverse scattering problem for cubic eingevalue problems of the class ${\varphi}_{xxx}+6q{\varphi}_{x}+6r\varphi =\lambda \varphi $,”
*Journal of Applied Mathematics and Mechanics*, vol. 52, pp. 361–365, 1988. View at: Google Scholar - A.-M. Wazwaz, “The Hirota's direct method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Ito seventh-order equation,”
*Applied Mathematics and Computation*, vol. 199, no. 1, pp. 133–138, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. Jafari, A. Yazdani, J. Vahidi, and D. D. Ganji, “Application of He's variational iteration method for solving seventh order Sawada-Kotera equations,”
*Applied Mathematical Sciences*, vol. 2, no. 9–12, pp. 471–477, 2008. View at: Google Scholar | MathSciNet - J. D. Cole, “On a quasi-linear parabolic equation occurring in aerodynamics,”
*Quarterly of Applied Mathematics*, vol. 9, pp. 225–236, 1951. View at: Google Scholar | MathSciNet - E. Hopf, “The partial differential equation ${u}_{t}+u{u}_{x}={u}_{xx}$,”
*Communications on Pure and Applied Mathematics*, vol. 3, pp. 201–230, 1950. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. M. Wazwaz, “The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations,”
*Applied Mathematics and Computation*, vol. 184, no. 2, pp. 1002–1014, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. H. Salas, “Some solutions for a type of generalized Sawada-Kotera equation,”
*Applied Mathematics and Computation*, vol. 196, no. 2, pp. 812–817, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. H. Salas, “Exact solutions for the general fifth KdV equation by the exp function method,”
*Applied Mathematics and Computation*, vol. 205, no. 1, pp. 291–297, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

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.