Solitary wave solutions of two nonlinear physical models by tanh–coth method

doi:10.1016/j.cnsns.2008.07.004

Communications in Nonlinear Science and Numerical Simulation

Volume 14, Issue 5, May 2009, Pages 1804-1809

https://doi.org/10.1016/j.cnsns.2008.07.004 Get rights and content

Abstract

In this work, we established the exact solutions for some nonlinear physical models. The tanh–coth method was used to construct solitary wave solutions of nonlinear evolution equations. The tanh–coth method presents a wider applicability for handling nonlinear wave equations.

Introduction

In the recent years, the investigation of the travelling wave solutions for nonlinear partial differential equations plays an important role in the study of nonlinear physical phenomena. Nonlinear wave phenomena appear in various scientific and engineering fields, such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, chemical physics and geochemistry. Nonlinear wave phenomena of dispersion, dissipation, diffusion, reaction and convection are very important in nonlinear wave equations. In the recent years, new exact solutions may help to find new phenomena. A variety of powerful methods such as inverse scattering method [1], [2], hirota bilinear method [3], [4], [5], the tanh method [6], [7], [8], [9], extended tanh method [10], [11], [12], sine–cosine method [13], [14], homogeneous balance method [15], [16], [17], $(\frac{G^{'}}{G})$ -expansion method [18], [19] and improved tanh-function [20] method were used to develop nonlinear dispersive and dissipative problems.

In the pioneer work of Malfliet [6], [7], the powerful tanh method was introduced for a reliable treatment of the nonlinear wave equations. The useful tanh method is widely used by many such as in [21], [22], [23] and by the references therein. Later, the tanh–coth method, developed by Wazwaz [24], [25], is a direct and effective algebraic method for handling nonlinear equations. Various extensions of the method were developed as well.

The next goal is the determination of new solitary wave solutions to highlight the power of the proposed method. The ease of using this method shows its power and its efficiency. The next interest is in the determination of exact travelling wave solutions for foam drainage and (2 + 1)-dimensional coupled nonlinear extension of the reaction–diffusion (CNLERD) equations. Searching for exact solutions of nonlinear problems has attracted a considerable amount of research work where computer symbolic systems facilitate the computational work.

Section snippets

The tanh–coth method

Wazwaz has summarized that by using tanh–coth method, a PDE $P (u, u_{t}, u_{x}, u_{xx}, \dots) = 0$ can be converted to an ODE $Q (U, U^{'}, U^{″}, U^{‴}, \dots) = 0$ upon using a wave variable $ξ = x - ct$ . Eq. (2.2) is then integrated as long as all terms contain derivatives where integration constants are considered zeros. Introducing a new independent variable $Y = \tanh (ξ), ξ = x - ct,$ leads to the change of derivatives: $\begin{matrix} \frac{d}{d ξ} = (1 - Y^{2}) \frac{d}{d Y}, \\ \frac{d^{2}}{d ξ^{2}} = (1 - Y^{2}) (- 2 Y \frac{d}{d Y} + (1 - Y^{2}) \frac{d^{2}}{d Y^{2}}), \\ \frac{d^{3}}{d ξ^{3}} = (1 - Y^{2}) ((6 Y^{2} - 2) \frac{d}{d Y} - 6 Y (1 - Y^{2}) \frac{d^{2}}{d Y^{2}} + (1 - Y^{2})^{2} \frac{d^{3}}{d Y^{3}}) . \end{matrix}$ The tanh–coth method [10], [11] admits

The foam drainage equation

Consider the foam drainage equation [26] $u_{t} + {(u^{2} - \frac{\sqrt{u}}{2} u_{x})}_{x} = 0,$ where x and t are scaled position and time coordinates, respectively. In this paper, we show the effectiveness and convenience of the method by obtaining the exact solution of Eq. (3.1). Foam is central to a number of everyday activities, both natural and industrial. As such foam has been of great interest for academic research. In the process industries, foam can be a desirable and even essential element of a process. An example is in the

The (2 + 1)-dimensional CNLERD equation

The (2 + 1)-dimensional CNLERD equation given by [29]is $\begin{matrix} u_{t} + u_{xy} - wu = 0, \\ v_{t} - v_{xy} + wv = 0, \\ w_{x} + (uv)_{y} = 0, \end{matrix}$ where $u, v$ and w are physical observables, and subscripts denote partial differentiation. Another physical application of Eq. (4.1) has been pointed out by Duan et al. [30] while presenting Eq. (4.1) as a corresponding geometric equivalent (2 + 1)-dimensional CNLERD equation of the integrable (2 + 1)-dimensional (modified) Heisenberg ferromagnet model. The complete integrability of this equation using the

Conclusion

The tanh–coth method was successfully used to establish solitary wave solutions. The obtained results complement the useful works of others for this important physical model. The proposed schemes are useful to be used in identical nonlinear dispersive models. Many well-known nonlinear wave equations were handled by this method to show the new solutions compared to the solutions obtained in [26], [28], [31]. The performance of the tanh–coth method is reliable and effective, and gives more

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Short communicationSolitary wave solutions of two nonlinear physical models by tanh–coth method

Abstract

Introduction

Section snippets

The tanh–coth method

The foam drainage equation

The (2 + 1)-dimensional CNLERD equation

Conclusion

Chaos Solitons Fract

Appl Math Comput

Appl Math Comput

Appl Math Comput

Commun Nonlinear Sci Numer Simul

Appl Math Comput

Appl Math Comput

Math Comput Modell

Commun Nonlinear Sci Numer Simul

Phys Lett A

Phys Lett A

Appl Math Comput

Phys Lett A

Phys Lett A

Nonlinear Anal

Short communication
Solitary wave solutions of two nonlinear physical models by tanh–coth method