A note on “The tanh–coth method combined with the Riccati equation for solving nonlinear coupled equation in mathematical physics”

Nikolai A. Kudryashov; Kirill E. Shilnikov

doi:10.1016/j.jksus.2012.06.001

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ORIGINAL ARTICLE

24 (

); 379-381

doi:

10.1016/j.jksus.2012.06.001

A note on “The tanh–coth method combined with the Riccati equation for solving nonlinear coupled equation in mathematical physics”

Nikolai A. Kudryashov^*, Kirill E. Shilnikov

Department of Applied Mathematics, National Research Nuclear University, “MEPhI”, 31 Kashirskoe Shosse, 115409 Moscow, Russian Federation

*Corresponding author. Tel.: +7 495 324 11 81. nakudr@gmail.com (Nikolai A. Kudryashov)

Received: 2012-5-20, Accepted: 2012-6-17, Published: 2012-10-01

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

Available online 26 June 2011

Peer review under responsibility of King Saud University.

Abstract

The recent paper “The tanh–coth method combined with the Riccati equation for solving nonlinear coupled equation in mathematical physics” (J. King Saud Univ. Sci. 23 (2011) 127–132) is analyzed. We show that the authors of this paper solved equations with the well known solutions. One of these equations is the famous Riccati equation and the other equation is one for the Weierstrass elliptic function. We present the general solutions of these equations. As this takes place, 19 solutions by authors do not satisfy the equation but the other 29 solutions can be obtained from the general solutions.

Keywords

Riccati equation

Weierstrass elliptic function

Tanh–coth method

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In the recent paper Bekir and Cevikel (2011) tried to study the (2+1)-dimensional system of two equations in the form

(1)

u_{t} + α u_{xxy} + 4 α {uv}_{x} + 4 α u_{x} v = 0,

(2)

u_{x} = v_{y}

However, in fact, these authors looked for exact solutions of these equations taking the traveling waves into account

(3)

u (x, t) = u (ξ), v (x, t) = v (ξ), ξ = x + y - β t

As a result using variables (3) the authors transform equations (1) and (2) to the system of nonlinear ordinary differential equations in the form:

(4)

- β u^{'} + α u^{‴} + 4 α {uv}^{'} + 4 α u^{'} v = 0

(5)

u^{'} = v^{'} .

After integrating both equations of system (4) and (5), the authors omitted constants of integration and obtained the system

(6)

α u^{″} - β u + 4 α u^{2} = 0,

(7)

u = v .

In fact, taking arbitrary constants into account we have the system of equations in the form

(8)

α u^{″} + 4 α u^{2} - Cu - C_{2} = 0

(9)

u = v - C_{1},

where C = β − 4αC₁ and C₁, C₂ are arbitrary constants. Authors used the tanh–coth method for finding some exact solutions of system (8) and (9). However the first equation of system (8) and (9) (or system (6) and (7)) has the well known general solution. Let us show this fact.

After multiplying on u′ Eq. (8) can be integrated with respect to ξ again. In this case we have the equation in the form

(10)

(u^{'})^{2} = - \frac{8}{3} u^{3} + D_{1} u^{2} + D_{2} u + D_{3},

where

D_{1} = \frac{C}{α}

D_{2} = \frac{2 C_{2}}{α}

and D₃ are arbitrary constants. The general solution of (10) was found more than one century ago and may be expressed via the Weierstrass elliptic function (see, for example, Polyanin and Zaitsev (2003); Kudryashov and Sinelshchikov (2012)). We can see it if we substitute

u = - \frac{3}{2} ℘ (ξ) + \frac{D_{1}}{8}

into Eq. (10). In this case we obtain the following equation for the Weierstrass elliptic function

(11)

\begin{matrix} (℘^{'})^{2} = 4 ℘^{3} - g_{2} ℘ - g_{3} \\ g_{2} = \frac{2 D_{2}}{3} + \frac{D_{1}^{2}}{12}, g_{3} = - \frac{4 D_{3}}{9} - \frac{D_{1}^{3}}{216} - \frac{D_{2} D_{1}}{18} \end{matrix}

As a result we have solution of Eq. (8) in the form

(12)

u = \frac{C}{8 α} - \frac{3}{2} ℘ (ξ - ξ_{0}, g_{2}, g_{3})

where

(13)

g_{2} = \frac{4 C_{2}}{3 α} + \frac{C^{2}}{12 α^{2}}, g_{3} = - \frac{4 D_{3}}{9} - \frac{C^{3}}{216 α^{3}} - \frac{C_{2} C}{9 α^{2}}

We checked all solutions by Bekir and Cevikel (2011) and obtained that 19 solutions from this paper: u₃, u₄, u₁₀, u₁₁, u₁₂, u₁₃, u₁₆, u₁₇, u₂₃, u₂₄, u₂₅, u₂₆, u₄₀, u₄₁, u₄₂, u₄₃, u₄₄, u₄₇ and u₄₈ do not satisfy Eq. (6) and sequently these solutions are wrong. The other 29 solutions of Eq. (6) of Bekir and Cevikel (2011) can be obtained from solution (12). For example, when D₂ and D₃ in (11) are equal to zero, solution of (6) can be transformed to the form:

(14)

u = - \frac{3 D_{1}}{8} + \frac{3 D_{1}}{8} \tanh^{2} (\sqrt{- \frac{D_{1}}{4}} (ξ - ξ_{0})) .

Assuming D₁ = −1 and ξ₀ = 0, we obtain

(15)

u = \frac{3}{8} (1 - \tanh^{2} (\frac{ξ}{2})) = \frac{3}{8} \frac{1}{\cosh^{2} (\frac{ξ}{2})} = \frac{3}{8} sec h^{2} (\frac{ξ}{2})

which coincides with u₂ of paper by Bekir and Cevikel (2011) (see formula (31)).

For D₁ = 4 we have

(16)

u = - \frac{3}{2} (1 + \tan^{2} (ξ - ξ_{0})) = - \frac{3}{2} \frac{1}{\cos^{2} (ξ - ξ_{0})} = - \frac{3}{2} \sec^{2} (ξ - ξ_{0})

For ξ₀ = 0 the obtained solution coincides with u₉ in work by Bekir and Cevikel (2011) (see formula (38)). If

ξ_{0} = \frac{π}{2}

our solution coincides with u₈ in Bekir and Cevikel (2011) (see formula (37)). So it is not hard to make sure that all 29 “new” solutions found by Bekir and Cevikel (2011) are contained in the solution (14) and consequently in general solution (12).

Bekir and Cevikel (2011) also considered the Riccati equation in the form

(17)

Y^{'} = A + BY + {CY}^{2},

where A, B and C are constants. Authors applied the tanh–coth method at B = 0 to look for exact solution of Eq. (17). However one can note that Eq. (17) also has the general solution. In the case B = 0 the general solution of Eq. (17) takes the form

(18)

Y = \sqrt{\frac{A}{C}} \tan (\sqrt{AC} (D + ξ)),

where D is an arbitrary constant. It is not hard to see that all 12 specific solutions (formulae (10) in Bekir and Cevikel (2011), which are used for constructing the solutions of Eq. (6), are contained in (18). For example, if we take

A = - C = \frac{1}{2}

we have

Y_{1} = \tanh (\frac{ξ}{2})

in the case D = 0 and we obtain

Y_{1} = \coth (\frac{ξ}{2})

in the case of D = −iπ. For A = 1, C = 4 we obtain both solutions Y₈ which differs only in the phase of the argument (i.e. arbitrary constant D).

In conclusion, let us note that idea by Bekir and Cevikel (2011) using the simplest equation method for finding exact solutions was published in paper by Kudryashov (1990, 2005). We have to point out that the authors of work Bekir and Cevikel (2011) made many mistakes discussed in papers Kudryashov (2009a,b), Kudryashov and Loguinova (2009), Kudryashov and Soukharev (2009), Kudryashov (2010), Kudryashov and Ryabov (2010), Kudryashov et al. (2010), Parkes (2009, 2010a,b,c).

Acknowledgements

This research was partially supported by Federal Target Programmes “Research and Scientific– Pedagogical Personnel of Innovation in Russian Federation on 2009–2013” and “Researches and developments in priority directions of development of a scientifically-technological complex of Russia on 2007–2013”.

References

Bekir A., Cevikel A.C., . The tanh–coth method combined with the Riccati equation for solving nonlinear coupled equation in mathematical physics. J. King Saud Univ. Sci.. 2011;23(2):127-132.
[Google Scholar]
Kudryashov N.A., . Exact solutions of the generalized Kuramoto–Sivashinsky equation. Phys. Lett. A. 1990;147(5–6):287-291.
[Google Scholar]
Kudryashov N.A., . Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos Soliton Fract.. 2005;24(5):1217-1231.
[Google Scholar]
Kudryashov N.A., . Seven common errors in finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simulat.. 2009;14(9–10):3503-3529.
[Google Scholar]
Kudryashov N.A., Loguinova N.B., . Be careful with Exp-function method. Commun. Nonlinear Sci. Numer. Simulat.. 2009;14(5):1881-1890.
[Google Scholar]
Kudryashov N.A., . On “new travelling wave solutions” of the KdV and the KdV-Burgers equations. Commun. Nonlinear Sci. Numer. Simulat.. 2009;14(5):1891-1900.
[Google Scholar]
Kudryashov N.A., Ryabov P.N., . Comment on: “Application of the G′/G method for the complex KdV equation [Huiqun Zhang, Commun Nonlinear Sci Numer Simul 15;2010:1700–1704] Commun. Nonlinear Sci. Numer. Simulat.. 2010;16(1):596-598.
[Google Scholar]
Kudryashov N.A., Ryabov P.N., Sinelshchikov D.I., . A note on “New kink-shaped solutions and periodic wave solutions for the (2+1)-dimensional Sine–Gordon equation. Appl. Math. Comput.. 2010;216(8):2479-2481.
[Google Scholar]
Kudryashov N.A., Sinelshchikov D.I., . Nonlinear differential equations of the second, third and fourth order with exact solutions. Appl. Math. Comput.. 2012;218(11):10454-10467.
[Google Scholar]
Kudryashov N.A., Soukharev M.B., . Popular ansatz methods and solitary wave solutions of the Kuramoto–Sivashinsky equation. Regul. Chaotic Dyn.. 2009;14(3):407-419.
[Google Scholar]
Kudryashov N.A., . Meromorphic solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simulat.. 2010;15(10):2778-2790.
[Google Scholar]
Parkes E.J., . A note on travelling – wave solutions to Lax's seventh-order KdV equation. Appl. Math. Comput.. 2009;215(2):864-865.
[Google Scholar]
Parkes E.J., . Observations on the tanh–coth expansion method for finding solutions to nonlinear evolution equations. Appl. Math. Comput.. 2010;217(4):1749-1754.
[Google Scholar]
Parkes E.J., . A note on solitary travelling-wave solutions to the transformed reduced Ostrovsky equation. Commun. Nonlinear Sci. Numer. Simulat.. 2010;15(10):2769-2771.
[Google Scholar]
Parkes E.J., . Observations on the basic (G′/G) – expansion method for finding solutions to nonlinear evolution equations. Appl. Math. Comput.. 2010;217(4):1759-1763.
[Google Scholar]
Polyanin A.D., Zaitsev V.F., . Handbook of Exact Solutions for Ordinary Differential Equations. Boca Raton: CRC Press; 2003.

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[1] Bekir A., Cevikel A.C., . The tanh–coth method combined with the Riccati equation for solving nonlinear coupled equation in mathematical physics. J. King Saud Univ. Sci.. 2011;23(2):127-132.
[Google Scholar]

[2] Kudryashov N.A., . Exact solutions of the generalized Kuramoto–Sivashinsky equation. Phys. Lett. A. 1990;147(5–6):287-291.
[Google Scholar]

[3] Kudryashov N.A., . Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos Soliton Fract.. 2005;24(5):1217-1231.
[Google Scholar]

[4] Kudryashov N.A., . Seven common errors in finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simulat.. 2009;14(9–10):3503-3529.
[Google Scholar]

[5] Kudryashov N.A., Loguinova N.B., . Be careful with Exp-function method. Commun. Nonlinear Sci. Numer. Simulat.. 2009;14(5):1881-1890.
[Google Scholar]

[6] Kudryashov N.A., . On “new travelling wave solutions” of the KdV and the KdV-Burgers equations. Commun. Nonlinear Sci. Numer. Simulat.. 2009;14(5):1891-1900.
[Google Scholar]

[7] Kudryashov N.A., Ryabov P.N., . Comment on: “Application of the G′/G method for the complex KdV equation [Huiqun Zhang, Commun Nonlinear Sci Numer Simul 15;2010:1700–1704] Commun. Nonlinear Sci. Numer. Simulat.. 2010;16(1):596-598.
[Google Scholar]

[8] Kudryashov N.A., Ryabov P.N., Sinelshchikov D.I., . A note on “New kink-shaped solutions and periodic wave solutions for the (2+1)-dimensional Sine–Gordon equation. Appl. Math. Comput.. 2010;216(8):2479-2481.
[Google Scholar]

[9] Kudryashov N.A., Sinelshchikov D.I., . Nonlinear differential equations of the second, third and fourth order with exact solutions. Appl. Math. Comput.. 2012;218(11):10454-10467.
[Google Scholar]

[10] Kudryashov N.A., Soukharev M.B., . Popular ansatz methods and solitary wave solutions of the Kuramoto–Sivashinsky equation. Regul. Chaotic Dyn.. 2009;14(3):407-419.
[Google Scholar]

[11] Kudryashov N.A., . Meromorphic solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simulat.. 2010;15(10):2778-2790.
[Google Scholar]

[12] Parkes E.J., . A note on travelling – wave solutions to Lax's seventh-order KdV equation. Appl. Math. Comput.. 2009;215(2):864-865.
[Google Scholar]

[13] Parkes E.J., . Observations on the tanh–coth expansion method for finding solutions to nonlinear evolution equations. Appl. Math. Comput.. 2010;217(4):1749-1754.
[Google Scholar]

[14] Parkes E.J., . A note on solitary travelling-wave solutions to the transformed reduced Ostrovsky equation. Commun. Nonlinear Sci. Numer. Simulat.. 2010;15(10):2769-2771.
[Google Scholar]

[15] Parkes E.J., . Observations on the basic (G′/G) – expansion method for finding solutions to nonlinear evolution equations. Appl. Math. Comput.. 2010;217(4):1759-1763.
[Google Scholar]

[16] Polyanin A.D., Zaitsev V.F., . Handbook of Exact Solutions for Ordinary Differential Equations. Boca Raton: CRC Press; 2003.