A note on “Jacobi elliptic function solutions for the modified Korteweg–de Vries equation”

Hong-Zhun Liu

doi:10.1016/j.jksus.2014.01.002

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SHORT COMMUNICATION

26 (

); 159-160

doi:

10.1016/j.jksus.2014.01.002

A note on “Jacobi elliptic function solutions for the modified Korteweg–de Vries equation”

Hong-Zhun Liu^*

Zhijiang College, Zhejiang University of Technology, Post-Office Box: 6044, Zhijiang Road 182, Hangzhou 310024, China

*Tel.: +86 571 87317741 mathlhz@163.com (Hong-Zhun Liu)

Received: 2013-10-1, Accepted: 2014-1-11, Published: 2014-04-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Available online 23 January 2014

Abstract

The recently published paper “Jacobi elliptic function solutions for the modified Korteweg–de Vries equation” [J. King Saud Univ. Sci. 25 (2013) 271–274] is analyzed. We show that these Jacobi elliptic function solutions obtained by the authors do not satisfy the original modified Korteweg–de Vries equation.

Keywords

Modified Korteweg–de Vries equation

Traveling wave solution

Tanh method

Common error

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Recently, Wang and Xiang (2013) studied the modified Korteweg–de Vries (mKdV) equation in the form

(1)

λ u_{t} - μ u^{2} u_{x} + {wu}_{xxx} = 0,

where

λ, μ

and w are constant parameters. They sought solutions by taking the traveling waves into account

(2)

u (x, t) = u (ξ), ξ = α x + kt .

As a result they transform (1) to an ordinary differential equation (ODE) in the form

(3)

c + k λ u - \frac{μ α}{3} u^{3} + w α^{3} u_{ξ ξ} = 0,

where c is an integral constant. They expressed u as

(4)

u = u_{0} + {pu}_{1} + p^{2} u_{2},

where p is a small parameter, and

u_{0}, u_{1}

and

u_{2}

are ansatz functions to be determined.

Substituting (4) into (3), we have

(5)

c + k λ u_{0} - \frac{μ α}{3} u_{0}^{3} + w α^{3} u_{0 ξ ξ} + p \cdot (k λ u_{1} - μ α u_{0}^{2} u_{1} + w α^{3} u_{1 ξ ξ}) + p^{2} \cdot (k λ u_{2} - μ α u_{0}^{2} u_{2} - μ α u_{0} u_{1}^{2} + w α^{3} u_{2 ξ ξ}) - p^{3} \cdot (2 μ α u_{0} u_{1} u_{2} + \frac{μ α}{3} u_{1}^{3}) - p^{4} \cdot μ α u_{2} (u_{0} u_{2} + u_{1}^{2}) - p^{5} \cdot μ α u_{1} u_{2}^{2} - p^{6} \cdot \frac{μ α}{3} u_{2}^{3} = 0 .

Since (4) is a solution of (3), (5) must hold for all values of p. Then each coefficient of p must vanish independently. Thus

(6)

c + k λ u_{0} - \frac{μ α}{3} u_{0}^{3} + w α^{3} u_{0 ξ ξ} = 0,

(7)

k λ u_{1} - μ α u_{0}^{2} u_{1} + w α^{3} u_{1 ξ ξ} = 0,

(8)

k λ u_{2} - μ α u_{0}^{2} u_{2} - μ α u_{0} u_{1}^{2} + w α^{3} u_{2 ξ ξ} = 0,

(9)

2 μ α u_{0} u_{1} u_{2} + \frac{μ α}{3} u_{1}^{3} = 0,

(10)

μ α u_{2} (u_{0} u_{2} + u_{1}^{2}) = 0, μ α u_{1} u_{2}^{2} = 0, \frac{μ α}{3} u_{2}^{3} = 0 .

The solution of (9) and (10) is $u_{2} = u_{1} = 0$ . In this case, the left hand side of (7) and (8) vanishes. So $u = u_{0}$ , and both of u and $u_{0}$ satisfy the same ordinary differential equation, which means nothing can be obtained from above approach.

It is a pity that Wang and Xiang (2013) obtained $u_{0}, u_{1}$ and $u_{2}$ through solving only three equations, namely (6)–(8) and neglected the rest of the equations. Then they claimed “The Jacobi elliptic function solutions, the trigonometric solutions and hyperbolic solutions are obtained”. It is not difficult to find that these solutions, namely (22)–(24) in their paper, are not admitted by the original mKdV equation.

In particular, the solution (24) in their paper, namely,

(11)

u (x, t) = a_{1} \tanh ξ + {pg}_{0} {sech}^{2} ξ + p^{2} b_{1} \tanh ξ + p^{2} b_{3} \tanh^{3} ξ

can also be checked by the tanh method (see, for example, Malfliet (2004)) easily. In fact, by replacing

{sech}^{2} ξ

with

1 - \tanh^{2} ξ

, the solution (11) is transformed into the following

(12)

u (x, t) = {pg}_{0} + (a_{1} + p^{2} b_{1}) Y - {pg}_{0} Y^{2} + p^{2} b_{3} Y^{3},

where

Y = \tanh ξ

is an introduced variable.

The tanh method proposes the following solution admitted by (3)

(13)

u = a_{0} + a_{1} Y + \dots + a_{N} Y^{N} = \sum_{n = 0}^{n = N} a_{n} Y^{n},

where N is a positive integer to be determined. Balancing the highest order linear term

u_{ξ ξ}

in (3) with the highest order nonlinear term

u^{3}

gives

N + 2 = 3 N

, so

N = 1

, which means the proposed solution will be

(14)

u = a_{0} + a_{1} Y .

However, the solution (12) indicates

N = 3

. This is a contradiction.

Finally, we notice that $c = 0$ in the results of Wang and Xiang (2013). And in this case, (3) can be changed to an ODE of degree four in the form

(15)

u_{ξ}^{2} = h_{0} + h_{2} u^{2} + h_{4} u^{4},

where

h_{0}

is an arbitrary constant, and

h_{2}

and

h_{4}

are certain constants. Many exact solutions of (15) have been presented and have played an important role in recently published papers, for example, Shang (2010); Alofi and Abdelkawy (2012); Ebaid and Aly (2012); Li et al. (2012); Ma et al. (2012); and Malik et al. (2012b,a). And its general solutions can be found in some literature, for example, Whittaker and Watson (1996) and Liu et al. (2014). Consequently, we can obtain general solutions of (3) in the case of

c = 0

directly. Due to limited space, here we omit its discussion.

Remark

It is worth to mention that if we regard the results obtained by Wang and Xiang (2013) as approximate solutions (in contrast to exact solutions), we can find that the derivations and results are correct. However, in this sense, these authors should express them in a proper way, and relevant calculated precisions for these solutions should be discussed as well, for example, see Holmes (2013).

In summary, solutions obtained by these authors do not satisfy the original mKdV equation. We have to point out that similar concerns are discussed in some other published papers as well (see, for examples, Kudryashov (2009) and Kudryashov and Shilnikov (2012)). We believe that our work will help people have a good understanding of the results obtained by Wang and Xiang (2013).

Acknowledgment

Many thanks are due to the helpful comments and suggestions from the anonymous referees and the editors.

References

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[1] Alofi A., Abdelkawy M., . New exact solutions of Boiti–Leon–Manna–Pempinelli equation using extended F-expansion method. Life Sci.. 2012;9:3395-4002.
[Google Scholar]

[2] Ebaid A., Aly E.H., . Exact solutions for the transformed reduced Ostrovsky equation via the F-expansion method in terms of Weierstrass-elliptic and Jacobian-elliptic functions. Wave Motion. 2012;49:296-308.
[Google Scholar]

[3] Holmes M.H., . Introduction to Perturbation Methods (2nd edition). New York: Springer; 2013.

[4] Kudryashov N.A., . Seven common errors in finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simulat.. 2009;14:3507-3529.
[Google Scholar]

[5] Kudryashov N.A., Shilnikov K.E., . A note on “the tanh–coth method combined with the riccati equation for solving nonlinear coupled equation in mathematical physics”. J. King Saud Univ. Sci.. 2012;24:379-381.
[Google Scholar]

[6] Li W.W., Tian Y., Zhang Z., . F-expansion method and its application for finding new exact solutions to the sine-Gordon and sinh-Gordon equations. Appl. Math. Computat.. 2012;219:1135-1143.
[Google Scholar]

[7] Liu H.Z., Sun X.Q., Chen L.J., . Comment on: “An observation on the periodic solutions to nonlinear physical models by means of the auxil-iary equation with a sixth-degree nonlinear term” [Commun Nonlinear SciNumer Simulat 2013;18:2177–2187] Commun Nonlinear Sci Numer Simulat. 2014;19:2553-2557.
[Google Scholar]

[8] Ma H.C., Zhang Z.P., Deng A.P., . A new periodic solution to Jacobi elliptic functions of MKdV equation and BBM equation. Acta Math. Appl. Sin. English Ser.. 2012;28:409-415.
[Google Scholar]

[9] Malfliet W., . The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations. J. Computat. Appl. Math. 2004:529-541.
[Google Scholar]

[10] Malik A., Chand F., Kumar H., Mishra S.C., . Exact solutions of nonlinear diffusion-reaction equations. Indian J. Phys.. 2012;86:129-136.
[Google Scholar]

[11] Malik A., Chand F., Kumar H., Mishra S.C., . Exact solutions of the Bogoyavlenskii equation using the multiple ( $\frac{G^{'}}{G}$ )-expansion method. Comput. Math. Appl.. 2012;64:2850-2859.
[Google Scholar]

[12] Shang D., . Exact solutions of coupled nonlinear Klein–Gordon equation. Appl. Math. Computat.. 2010;217:1577-1583.
[Google Scholar]

[13] Wang H., Xiang C., . Jacobi elliptic function solutions for the modified Korteweg–de Vries equation. J. King Saud Univ. Sci.. 2013;25:271-274.
[Google Scholar]

[14] Whittaker E.T., Watson G.N., . A Course of Modern Analysis (4th ed.). Cambridge: Cambridge University Press; 1996.