Comment on “New Exact Solutions to the KdV-Burgers-Kuramoto Equation with the Exp-Function Method”

Abstract

We point out in this paper that the claims made by Kim et al. in the commented paper are incorrect and no new exact solution was obtained.

1. Introduction

In [1], Kim and Chun had investigated exact solutions to the following KdV-Burgers-Kuramoto equation: They took traveling wave transformation with into account and transformed (1) into an ordinary differential equation: After implementing the Exp-function method [2] based on the truncated Painlevé, they had constructed the following four new generalized solitary wave solutions to (1).

Case 1. Consider the following: where , as in the following, is an introduced variable, and , , , and are arbitrary constants.

Other Three Cases. Consider where and , are arbitrary constants.

Case 2. , is an arbitrary constant, and subjects to

Case 3. , and and subject to

Case 4. , and and subject to The authors claimed that the above four solutions could not be directly constructed form the Exp-function method.

They also claimed that they had obtained a new solitary wave solution in the case where and , which was given by

2. Comment and Analysis

In this section, we will analyze the claims by Kim and Chun in [1].

2.1. Comment 1

It is not difficult to rewrite (3) and (5) in the following form: where and are certain constants.

Therefore, we have According to the Exp-function method [2], we can reobtain the solution (11) by assuming that the solution of (2) can be expressed in the form where (i) and , (ii) and , (iii) , respectively. It is worth to mention that the fact that the three cases of (i), (ii), and (iii) are equivalent has been emphasized in [3–5]. Thus the claim in the commented paper that “new generalized solitary wave solutions are constructed for the KdV-Burgers-Kuramoto equation, which cannot be directly constructed from the Exp-function method” is not true.

2.2. Comment 2

In this section, we show that the solutions in the mentioned four cases are incorrect. Here, we should point out that it is difficult for us to solve original algebra system appearing in [1] and therefore we verify the four cases in an ad hoc way.

2.2.1. Solution Analysis

At the beginning, substituting (4) into (3), we find and substituting (6) into (5), we find

So we assume that the solution of (2) can be expressed in the form where , , and are constants to be determined.

We emphasize that (16) needs only three undetermined parameters. Unlike (5), there are ten undetermined parameters. Hence assumption (16) can reduce the computation burden.

Substituting (16) into (2) and setting the coefficients of all powers of to zero yield a system of algebraic equations for , , , , and . Solving these algebraic equations, we can determine certain solutions to (2). Obviously, these solutions cover (14) and (15). However, it is to our surprise that with the aid of Maple we determine none of nontrivial solutions after solving the above system equations. Hence further verification should be made.

2.2.2. Solution Check

In what follows, after careful numerical inspection, we show that the four cases are incorrect.

Firstly, we check the solution in Case 1, namely, (3) and (4) (or (14)) with arbitrary constants , , , and . We observe that the nontrivial solution (3) with (4) is independent of , which is impossible. Indeed, given is a nontrivial solution of Case 1, we rewrite (2) as Substituting into (17), we have Fixing , , and and leaving free, we can find that the right-hand side of (18) is determined, while the left-hand side is not. This is a contradiction.

Secondly, we take Case 2, namely, (5) with (6) and (7), into account. Setting , , , , and for simplicity, we have Substituting above values into the left-hand side of (2), we obtain where And by taking , we have Since the right-hand side of (20) is not zero for all value of , we conclude that the solution in Case 2 is not admitted by the original ordinary differential equation (2) and KdV-Burgers-Kuramoto equation (1).

Case 3 and Case 4 can be checked in a way similar to Case 2; here we omit the details.

At the end of this section, we should point out that (10) can be exactly simplified to the constant as follows: which is trivial.

So, we conclude that not any new exact solution was obtained.

3. Conclusion

In this paper, we emphasize that the paper [1] contains some errors. We have to point out that similar mistakes had been analyzed in some published papers (see, e.g., [6, 7]). We hope that the results will help people have a good understanding of the work made by Kim et al.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by NSF of China (11201427).