Jacobi elliptic function solutions for the modified Korteweg–de Vries equation

Honglei Wang; Chunhuan Xiang

doi:10.1016/j.jksus.2013.04.001

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SHORT COMMUNICATION

25 (

); 271-274

doi:

10.1016/j.jksus.2013.04.001

Jacobi elliptic function solutions for the modified Korteweg–de Vries equation

Honglei Wang^{^a,*}, Chunhuan Xiang^{^b}

a Faculty of basic medical science, Chongqing Medical University, Chongqing 400016, PR China

b College of Mathematics and Statistics, Chongqing University of Arts and Sciences, Yongchuan, Chongqing 402160, PR China

*Corresponding author tianteradar@yahoo.com.cn (Honglei Wang)

Received: 2013-3-19, Accepted: 2013-4-18, Published: 2013-07-01

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 29 April 2013

Abstract

Formula solutions to the modified Korteweg–de Vries (mKdV) equation with constant coefficients are obtained via the Jacobi elliptic periodic function transform method and symbolic computation. Those periodic solutions degenerate as the corresponding hyperbolic function solutions when the modulus is m → 1 and trigonal solutions with m → 0.

Keywords

Modified Korteweg–de Vries equation

Jacobi elliptic function

Trigonal solutions

Hyperbolic solutions

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1 Introduction

The investigation of nonlinear partial differential equations plays an important role in the study of nonlinear physical phenomena. A variety of powerful and direct methods have been developed in this direction (Bhrawy et al., 2013; Ebadi et al., 2011, 2012, 2013; Krishnan et al., 2011, 2012; Biswas et al., 2012, 2013 ). In this Letter, the modified Korteweg–de Vries (mKdV) equation is investigated, which has been mentioned in many branches of nonlinear science field, and given in the form

(1)

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

where the subscripts denote the partial derivates of x and t, λ, μ and w are constant parameters, u represents a real scalar function u(x,t). The equation and equations derived from itself have been widely studied (Trogdon et al., 2012; Bozhkov et al., 2013; Salkuyeh and Bastani, 2013; Ono, 1992; Liang et al., 2011; Anco et al., 2011 ) because these equations play an important role in many physics contexts, such as anharmonic lattices (Ono, 1992), ion acoustic solitons (Konno and Ichikawa, 1974; Watanabe, 1984; Lonngren, 1998 ), traffic jam (Komatsu and Sasa, 1995; Nagatani, 1999), thin ocean jets (Cushman-Roisin et al., 1992; Ralph and Pratt, 1994), internal waves (Grimshaw et al., 2004; Grimshaw, 2001), heat pulses in solids (Tappert and Varma, 1970), and so on. With the help of an auxiliary Lamé equation (Fu et al., 2009; Liu, 2009), the Jacobi elliptic function solutions, which are important to label the unidentified spectrum from white dwarf stars (10²–10⁵T) and neutron stars(10⁷–10⁹T), the periodic solutions for mKdV equations are shown.

2 The Jacobi elliptic function and its properties

Usually, three types of Jacobi elliptic functions (Bhrawy et al., 2013) (sn(ξ,m), cn(ξ,m), dn(ξ,m)) have the following properties: $\begin{matrix} cn (ξ, m)^{2} + sn (ξ, m)^{2} = 1; \\ dn (ξ, m)^{2} + m^{2} sn (ξ, m)^{2} = 1; \end{matrix}$ the derivates of Jacobi elliptic functions can be expressed as

(2)

\begin{matrix} \frac{dcn (ξ, m)}{d ξ} = - sn (ξ, m) dn (ξ, m), \\ \frac{ddn (ξ, m)}{d ξ} = - m^{2} sn (ξ, m) cn (ξ, m), \\ \frac{dsn (ξ, m)}{d ξ} = cn (ξ, m) dn (ξ, m); \end{matrix}

where m is the modulus of Jacobi elliptic functions. The Jacobi elliptic functions will asymptotically go into hyperbolic functions and trigonometric functions when the modulus is m → 1 and m → 0, respectively.

\begin{matrix} m \to 1, sn (ξ, m) \to tanh (ξ), cn (ξ, m) \to sech (ξ), \\ dn (ξ, m) \to sech (ξ); \\ m \to 0, sn (ξ, m) \to sin (ξ), cn (ξ, m) \to cos (ξ), dn (ξ, m) \to 1 . \end{matrix}

The Lamé equation (Fu et al., 2009; Liu, 2009) in terms of φ(ξ) can be expressed as

(3)

\frac{d^{2} φ}{d ξ^{2}} + (η - l (l + 1) m^{2} sn (ξ, m)^{2}) φ = 0;

where η is the eigenvalue of φ and l is a positive parameter. The particular solution of lamé equation can be written as:

when η = 4 + m² and l = 2,

(4)

φ = sn (ξ, m) cn (ξ, m);

when η = 1 + 4m² and l = 2,

(5)

φ = sn (ξ, m) dn (ξ, m);

when η = 1 + m² and l = 2,

(6)

φ = cn (ξ, m) dn (ξ, m) .

3 The Jacobi elliptic function solutions for mKdV equation

We will seek the solutions with one travelling wave-like variable ξ for mKdV equation: ξ = αx + k t, where α and k are constant parameters. Then, the evolution Eq. (1) can be transformed as follows :

(7)

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

Integrating Eq. (7) once with respect to ξ, we have

(8)

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

where c is the integral constant.

Usually, the scalar function u in terms of perturbation method can be expressed as

(9)

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

and the second derivative of Eq. (9) reads

(10)

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

where p(0 < p ≪ 1) is a small parameter, u₀, u₁ and u₂ are the zeroth-order, first-order and second-order ansatz solutions, respectively.

Substituting Eqs. (9) and (10) into Eq. (8) and equating the coefficients of p⁰, p¹ and p² to zero, we obtain the zeroth-order, first-order and second-order solutions of Eq. (7) in the following form:

(11)

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

(12)

{ku}_{1} λ - α μ u_{0}^{2} u 1 + w α^{3} u_{1 ξ ξ} = 0;

(13)

{ku}_{2} λ - α μ u_{0} u_{1}^{2} - α μ u_{0}^{2} u_{2} + w α^{3} u_{2 ξ ξ} = 0 .

The zeroth-order ansatz solution u₀ in terms of the Jacobi elliptic sine function reads

(14)

u_{0} = a_{0} + a_{1} sn (ξ, m) + a_{2} sn (ξ, m)^{2},

where a₀, a₁ and a₂ are constant parameters. By substituting Eq. (14) into Eq. (11) and collecting all terms with the same power of sn(ξ,m) together and equating the coefficients of the polynomials to zero, yield a set of simultaneous algebraic equations. Solving the algebraic equations above yields

k λ = w α^{3} (1 + m^{2}); a_{0} = 0; a_{2} = 0; c = 0; a_{1} = \pm \sqrt{\frac{6 {wm}^{2} α^{2}}{μ}} .

The zeroth-order solution u₀ in terms of kλ = wα³ (1 + m²) is of the form

(15)

u_{0} (ξ) = a_{1} sn (ξ, m) .

Substituting Eq. (15) into Eq. (12), we obtain

(16)

u_{1 ξ ξ} + (6 dn (ξ, m)^{2} - (6 - \frac{k λ}{w α^{3}})) u_{1} = 0 .

With the help of the Jacobi elliptic properties and Lamé equation Eq. (3), the solution of above equation is expressed as

(17)

u_{1} (ξ) = g_{0} cn (ξ, m) dn (ξ, m),

where g₀ is a constant parameter.

Using Eqs. 15 and 17, Eq. 13 is rewritten as

(18)

w α^{3} u_{2 ξ ξ} + (k λ - 6 {wm}^{2} α^{3} sn (ξ, m)^{2}) u_{2} - a_{1} sn (ξ, m) u_{1}^{2} α μ = 0,

The second-order ansatz solution u₂ is supposed to be

(19)

u_{2} = b_{0} + b_{1} sn (ξ, m) + b_{3} sn (ξ, m)^{3},

where b₀, b₁ and b₂ are constant parameters to be determined.

Substituting Eq. (19) into Eq. (18), we have a set of algebraic equations about sn(ξ,m) and obtain

(20)

b_{0} = 0; b_{1} = \frac{- g_{0}^{2} \sqrt{μ} (1 + m^{2})}{2 m α \sqrt{6 w}}; b_{3} = \frac{{mg}_{0}^{2} \sqrt{μ}}{α \sqrt{6 w}}

when

a_{1} = \sqrt{6 {wm}^{2} α^{2} / μ};

and

(21)

b_{0} = 0; b_{1} = \frac{- 7 g_{0}^{2} \sqrt{μ} (1 + m^{2})}{2 m α \sqrt{6 w}}; b_{3} = \frac{- {mg}_{0}^{2} \sqrt{μ}}{α \sqrt{6 w}},

when

a_{1} = - \sqrt{6 {wm}^{2} α^{2} / μ} .

The solution of Eq. (1) can be written as:

(22)

u (x, t) = a_{1} sn (ξ, m) + {pg}_{0} cn (ξ, m) dn (ξ, m) + p^{2} b_{1} sn (ξ, m) + p^{2} b_{3} sn (ξ, m)^{3} .

By using the Jacobi elliptic function properties, when the modulus is m → 1 and m → 0, the above equation is transformed into the following

(23)

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

(24)

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

where ξ = αx + kt. Eq. (23) is trigonometric solutions of Eq. (1), Eq. (24) is hyperbolic solutions of Eq. (1). The numerical simulations with

a_{1} = - \sqrt{6 {wm}^{2} α^{2} / μ}

are attached, all the figures are obtained in the contour plot form with the aid of Origin software (see Figs. 1–3 ).

Simulation u(x,t) for Eq. (22) with the modulus m = 0.4 , α = 0.2 , w = 2.5 , g 0 = 1 , a 1 = - 6 wm 2 α 2 / μ , μ = 1 , λ = 1 , k = w α 3 ( 1 + m 2 ) and p = 0.001, the range of x and t are x ∈ [−20,20] and t ∈ [−20,20], respectively.

simulation u(x,t) for Eq. (22) with the modulus m = 0.00000001 , α = 0.2 , w = 2.5 , g 0 = 1 , a 1 = - 6 wm 2 α 2 / μ , μ = 1 , λ = 1 , k = w α 3 ( 1 + m 2 ) and p = 0.001, the range of x and t are x ∈ [−20,20] and t ∈ [−20,20], respectively.

simulation u(x,t) for Eq. (22) with the modulus m = 0.99999999 , α = 0.2 , w = 2.5 , g 0 = 1 , a 1 = - 6 wm 2 α 2 / μ , μ = 1 , λ = 1 , k = w α 3 ( 1 + m 2 ) and p = 0.001, the range of x and t are x ∈ [−20,20] and t ∈ [−20,20], respectively.

4 Discussions and conclusions

In this paper, we presented the Jacobi elliptic function method for solving the mKdV equations. The Jacobi elliptic function solutions, the trigonometric solutions and hyperbolic solutions are obtained. This present work affirms that the Jacobi elliptic function method is an easy straight forward method to solve nonlinear partial differential equations.

Acknowledgement

This work was financially supported by the Science Foundation of Chongqing City Board of Education (KJ121206)

References

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Watanabe S., . Ion acoustic soliton in plasma with negative ion. Journal of the Physical Society of Japan. 1984;53:950-956.
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Introduction

The Jacobi elliptic function and its properties

The Jacobi elliptic function solutions for mKdV equation

Discussions and conclusions

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[1] Anco, S.C., Ngatat, N.T., Willoughby M., 2011. Interaction properties of complex mKdV solitons. arXiv:1102.3620 [nlin.SI].

[2] Bhrawy A.H., Biswas A., Javidi M., Ma W.-X., Pinar Z., Yildirim A., . New solutions for (1 + 1)-dimensional and (2 + 1)-dimensional Kaup–Kuperschmidt equations. Results in Mathematics. 2013;63:675-686.
[Google Scholar]

[3] Bhrawy A.H., Abdelkawy M.A., Biswas A., . Cnoidal and snoidal wave solutions to coupled nonlinear wave equations by the extended Jacobi’s elliptic function method. Communications in Nonlinear Science and Numerical Simulation. 2013;18(4):915-925.
[Google Scholar]

[4] Biswas A., Ebadi G., Fessak M., Johnpillai A.G., Johnson S., Krishnan E.V., Yildirim A., . Solutions of the perturbed Klein–Gordon equations. Iranian Journal of Science and Technology, Transaction A. 2012;36(A4):431-452.
[Google Scholar]

[5] Biswas A., Krishnan E.V., Suarez P., Kara A.H., Kumar S., . Solitary waves and conservation law of Bona–Chen equation. Indian Journal of Physics. 2013;87:169-175.
[Google Scholar]

[6] Bozhkov Y., Dimas S., Ibragimov N.H., . Conservation laws for a coupled variable-coefficient modified Korteweg–de Vries system in a two-layer fluid model. Communications in Nonlinear Science and Numerical Simulation. 2013;18:1127-1135.
[Google Scholar]

[7] Cushman-Roisin B., Pratt L.J., Ralph E.A., . A general theory for equivalent barotropic thin jets. Journal of Physical Oceangraphy. 1992;23:91-103.
[Google Scholar]

[8] Ebadi G., Krishnan E.V., Labidi M., Zerrad E., Biswas A., . Analytical and numerical solutions for Davey–Stewartson equation with power law nonlinearity. Waves in Random and Complex Media. 2011;21:559-590.
[Google Scholar]

[9] Ebadi G., Krishnan E.V., Biswas A., . Solitons and cnoidal waves of the Klein–Gordon Zakharov equation. Pramana. 2012;79:185-198.
[Google Scholar]

[10] Ebadi G., Fard N.Y., Bhrawy A.H., Kumar S., Triki H., Yildirim A., Biswas A., . Solitons and other solutions to (3 + 1)-dimensional extended Kadomtsev–Petviashvili equation with power law nonlinearity. Romanian Reports in Physics. 2013;65:27-62.
[Google Scholar]

[11] Fu Z., Yuan N., Mao J., Liu S., . Multi-order exact solutions to the Drinfeld–Sokolov–Wilson equations. Physics Letters A. 2009;373:3710-3714.
[Google Scholar]

[12] Grimshaw R., . Internal solitary waves. In: Grimshaw R., ed. Environmental Stratified Flows. Boston: Kluwer Academic; 2001. p. :1-27.
[Google Scholar]

[13] Grimshaw R., Pelinovsky E., Talipova T., Kurkin A., . Simulation of the transformation of internal solitary waves on oceanic shelves. Journal of Physical Oceangraphy. 2004;34:2774-2779.
[Google Scholar]

[14] Komatsu T.S., Sasa S.I., . Kink soliton characterizing traffic congestion. Physical Review E. 1995;52:5574-5582.
[Google Scholar]

[15] Konno K., Ichikawa Y.H., . A modified Korteweg–de Vries equation for ion acoustic waves. Journal of the Physical Society of Japan. 1974;37:1631-1636.
[Google Scholar]

[16] Krishnan E.V., Triki H., Labidi M., Biswas A., . A study of shallow water waves with Gardner’s equation. Nonlinear Dynamics. 2011;66:497-507.
[Google Scholar]

[17] Krishnan E.V., Kumar S., Biswas A., . Solitons and other nonlinear waves of the Boussinesq equation. Nonlinear Dynamics. 2012;70:1213-1221.
[Google Scholar]

[18] Liang Y., Wei G., Li X., . New variable separation solutions and nonlinear phenomena for the (2 + 1)-dimensional modified Korteweg–de Vries equation. Communications in Nonlinear Science and Numerical Simulation. 2011;16:603-609.
[Google Scholar]

[19] Liu G.-T., . Lame equation and asymptotic higher-order periodic solutions to nonlinear evolution equations. Applied Mathematics and Computation. 2009;212:312-317.
[Google Scholar]

[20] Lonngren K.E., . Ion acoustic soliton experiment in a plasma. Optical and Quantum Electronics. 1998;30:615-630.
[Google Scholar]

[21] Nagatani T., . TDGL and mKdV equations for jamming transition in the lattice models of traffic. Physica A. 1999;264:581-592.
[Google Scholar]

[22] Ono H., . Soliton fission in anharmonic lattices with reflectionless inhomogeneity. Journal of the Physical Society of Japan. 1992;61:4336-4343.
[Google Scholar]

[23] Ralph E.A., Pratt L., . Predicting eddy detachment for an equivalent barotropic thin jet. Journal of Nonlinear Science. 1994;4:355-374.
[Google Scholar]

[24] Salkuyeh D.K., Bastani M., . Solution of the complex modified Korteweg–de Vries equation by the projected differential transform method. Applied Mathematics and Computation. 2013;219:5105-5112.
[Google Scholar]

[25] Tappert F.D., Varma C.M., . Asymptotic theory of self-trapping of heat pulses in solids. Physical Review Letters. 1970;25:1108-1111.
[Google Scholar]

[26] Trogdon T., Olver S., Deconinck B., . Numerical inverse scattering for the Korteweg–de Vries and modified Korteweg–de Vries equations. Physica D: Nonlinear Phenomena. 2012;241:1003-1025.
[Google Scholar]

[27] Watanabe S., . Ion acoustic soliton in plasma with negative ion. Journal of the Physical Society of Japan. 1984;53:950-956.
[Google Scholar]