The extended tanh method for abundant solitary wave solutions of nonlinear wave equations

doi:10.1016/j.amc.2006.09.013

Applied Mathematics and Computation

Volume 187, Issue 2, 15 April 2007, Pages 1131-1142

https://doi.org/10.1016/j.amc.2006.09.013 Get rights and content

Abstract

The extended tanh method is used to establish abundant solitary wave solutions of nonlinear wave equations. The obtained solutions include solitons, kinks and plane periodic solutions. The study is an extension to the remarkable development by Malfliet [1]. The extended tanh method presents a wider applicability for handling nonlinear wave equations.

Introduction

Nonlinear wave equations have a significant role in several scientific and engineering fields. These equations appear in solid state physics, fluid mechanics, chemical kinetics, plasma physics, population models, nonlinear optics, propagation of fluxons in Josephson junctions, and many others. The pioneer work of Malfliet in [1] introduced the powerful hyperbolic tangent (tanh) method for a reliable treatment of the nonlinear wave equations. The useful tanh method is widely used by many such as in [2], [3], [4], [5], [6], [7], [8], [9], [10], [11] and by the references therein. The method introduces a unifying method that one can find exact as well as approximate solutions in a straightforward and systematic way [1], [2], [3], [4].

The tanh method has been subjected to many modifications that mainly depend on the Riccati equation and the solutions of well-known equations. The standard tanh method and the proposed modifications all depend on the balance method, where the linear terms of highest order are balanced with the highest order nonlinear terms of the reduced equation.

It is the aim of this work to further complement the studies of [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. It is interesting to point out that in these works, only one solitary wave solution was obtained for each investigated equation. We aim to use the extended tanh method [12], [13] to obtain many solutions to these equations instead of one. We also want to confirm the wider applicability of the extended tanh method. In what follows, we highlight the main steps of these two methods briefly. More details can be found in [14], [15], [16], [17], [18], [19], [20], [21], [22] and the references therein.

Section snippets

The methods

A PDE $P (u, u_{t}, u_{x}, u_{xx}, u_{xxx}, \dots) = 0$ can be converted to an ODE $Q (u, u^{'}, u^{″}, u^{‴}, \dots) = 0,$ upon using a wave variable ξ = (x − ct). Eq. (2) is then integrated as long as all terms contain derivatives where integration constants are considered zeros.

The Burgers equation

The Burgers equation is characterized by a dissipative term and models fluid turbulence. It reads $u_{t} + {uu}_{x} - u_{xx} = 0 .$ Using the wave variable ξ = x − ct carries Eq. (7) into the ODE $- cu + \frac{1}{2} u^{2} - u^{'} = 0$ obtained after integrating the ODE once and setting the constant of integration equal to zero. Balancing uu′ with u² in (8) gives $M + 1 = 2 M,$ so that $M = 1 .$ The extended tanh method (6) admits the use of the finite expansion $u (ξ) = a_{0} + a_{1} Y + \frac{b_{1}}{Y} .$ Substituting (11) into (8), and collecting the coefficients of Y we obtain a system of

The KdV equation

The KdV equation is characterized by the convection term uu_x and the dispersion term u_xxx. It reads $u_{t} + 6 {uu}_{x} + u_{xxx} = 0 .$ Using the wave variable ξ = x − ct carries Eq. (18) into the ODE $- cu + 3 u^{2} + u^{″} = 0$ obtained after integrating the ODE once and setting the constant of integration equal to zero. Balancing u″ with u² in (19) gives $M + 2 = 2 M,$ so that $M = 2 .$ The extended tanh method (6) admits the use of the finite expansion $u (ξ) = a_{0} + a_{1} Y + a_{2} Y^{2} + \frac{b_{1}}{Y} + \frac{b_{2}}{Y^{2}} .$ Substituting (22) into (19), collecting the coefficients of Y we obtain

The mKdV equation

The modified KdV (mKdV) equation serves as a model of solitons in a multicomponent plasma and phonons in anharmonic lattice [1]. The mKdV reads $u_{t} + 6 u^{2} u_{x} + u_{xxx} = 0 .$ Proceeding as before, the wave variable ξ = x − ct carries Eq. (41) into the ODE $- cu + 2 u^{3} + u^{″} = 0 .$ Balancing u″ with u³ in (42) gives $M + 2 = 3 M,$ so that $M = 1 .$

It is interesting to point out that in [1], it is stated that for M = 1 the tanh method does not lead to a real solution. Similarly, the extended tanh method gives only complex solutions. Moreover, a

The Fisher equation

The Fisher equation describes the process of interaction between diffusion and reaction. This equation is encountered in chemical kinetics and population dynamics which includes problems such as nonlinear evolution of a population in one-dimensional habitual, neutron population in a nuclear reaction [1]. It reads $u_{t} - u_{xx} - u (1 - u) = 0 .$ Using the wave variable ξ = x − ct carries Eq. (47) into the ODE $- {cu}^{'} - u^{″} - u (1 - u) = 0 .$ Balancing u″ with u² in (48) gives $M + 2 = 2 M,$ so that $M = 2 .$ The extended tanh method (6) applies the

The Burgers–Fisher equation

The Burgers–Fisher equation reads $u_{t} + {uu}_{x} + u_{xx} + u (1 - u) = 0,$ that will be carried into the ODE $- {cu}^{'} + {uu}^{'} + u^{″} + u (1 - u) = 0$ obtained after using the wave variable ξ = x − ct. Balancing uu′ with u″ gives $M + 2 = 2 M + 1,$ so that $M = 1 .$ Accordingly, the extended tanh method (6) is of the form $u (ξ) = a_{0} + a_{1} Y + \frac{b_{1}}{Y} .$ Substituting (62) into (59), and proceeding as before we obtain the three sets of solutions

(i)
The first set: $a_{0} = \frac{1}{2}, a_{1} = \frac{1}{2}, b_{1} = 0, c = \frac{5}{2}, μ = \frac{1}{4} .$
(ii)
The second set: $a_{0} = \frac{1}{2}, a_{1} = 0, b_{1} = \frac{1}{2}, c = \frac{5}{2}, μ = \frac{1}{4} .$
(iii)
The third set: $a_{0} = \frac{1}{2}, a_{1} = \frac{1}{4}, b_{1} = \frac{1}{4}, c = \frac{5}{2}, μ = \frac{1}{8} .$

This in

The Huxley equation

The Huxley equation reads $u_{t} - {au}_{xx} - u (k - u^{n}) (u^{n} - 1) = 0, n > 1 .$ This equation is used for nerve propagation in neurophysics and wall propagation in liquid crystals [10]. The wave variable ξ = x − ct carries out Eq. (69) into the ODE $- {cu}^{'} - {au}^{″} - (k + 1) u^{n + 1} + u^{2 n + 1} + ku = 0 .$ Balancing u″ with u²ⁿ⁺¹ gives $M + 2 = (2 n + 1) M,$ so that $M = \frac{1}{n} .$ To obtain a closed form solution, M should be an integer, therefore we use the transformation $u (x, t) = v^{\frac{1}{n}}$ and as a result Eq. (70) becomes $- {cnvv}^{'} - {anvv}^{″} - a (1 - n) (v^{'})^{2} - (k + 1) n^{2} v^{3} + n^{2} v^{4} + {kn}^{2} v^{2} = 0 .$ Balancing vv″

The nonlinear reaction–diffusion equations

The nonlinear reaction–diffusion equation appear in many forms [1], [10]. In this section we will focus our work on two of these reaction–diffusion equations, namely $u_{t} - (u^{2})_{xx} - u (1 - u) = 0$ and $u_{t} - (u^{3})_{xx} - u (1 - u^{2}) = 0 .$

Concluding remarks

The extended tanh method [12], [13] was successfully used to establish abundant solitary wave solutions, mostly solitons and kinks solutions. Many well known nonlinear wave equations were handled by this method to show the new solutions compared to the solutions obtained in [1], [2], [3], [4]. The performance of the extended tanh method is reliable and effective and gives more solutions. The applied method will be used in further works to establish more entirely new solutions for other kinds of

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The extended tanh method for abundant solitary wave solutions of nonlinear wave equations

Abstract

Introduction

Section snippets

The methods

The Burgers equation

The KdV equation

The mKdV equation

The Fisher equation

The Burgers–Fisher equation

The Huxley equation

The nonlinear reaction–diffusion equations

Concluding remarks

Appl. Math. Comput.

Appl. Math. Comput.

Comm. Nonlinear Sci. Numer. Simul.

Appl. Math. Comput.

Chaos, Solitons Fractals

Appl. Math. Comput.

Appl. Math. Comput.

Phys. Lett. A

Appl. Math. Comput.

Math. Comput. Modell.

Comput. Math. Appl.