About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 890574, 10 pages
http://dx.doi.org/10.1155/2012/890574
Research Article

Travelling Wave Solutions to the Benney-Luke and the Higher-Order Improved Boussinesq Equations of Sobolev Type

Department of Mathematics, Sakarya University, Sakarya, Turkey

Received 12 September 2012; Accepted 28 November 2012

Academic Editor: Dragoş-Pătru Covei

Copyright © 2012 Ömer Faruk Gözükızıl and Şamil Akçağıl. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

By using the tanh-coth method, we obtained some travelling wave solutions of two well-known nonlinear Sobolev type partial differential equations, namely, the Benney-Luke equation and the higher-order improved Boussinesq equation. We show that the tanh-coth method is a useful, reliable, and concise method to solve these types of equations.

1. Introduction

The term “Sobolev equation” is used in the Russian literature to refer to any equation with spatial derivatives on the highest order time derivative [1]. In other words, they are characterized by having mixed time and space derivatives appearing in the highest-order terms of the equation and were studied by Sobolev [2]. Equations of Sobolev type describe many physical phenomena [37]. In recent years considerable attention has been paid to the study of equations of Sobolev type. For more details we refer the reader to [8] and references therein.

The Benney-Luke equation is as follows: where and are positive numbers, such that is a Sobolev type equation and studied for a very long time. The dimensionless parameter is named the Bond number, which captures the effects of surface tension and gravity force and is a formally valid approximation for describing two-way water wave propagation in the presence of surface tension [9]. In [10] Pego and Quintero studied the propagation of long water waves with small amplitude. They showed that in the presence of a surface tension, the propagation of such waves is governed by (1.1), originally derived by Benney and Luke [11]. There are many studies concerning with this equation. Amongst them the stability analysis [9, 12], Cauchy problem [1315], existence and analyticity of solutions [16], and travelling wave solutions [17] can be mentioned.

In [18], Schneider and Wayne showed that in the longwave limit the water wave problem without surface tension can be described approximately by two decoupled KdV equations. They considered a class of Boussinesq equation which models the water wave problem with surface tension as follows: where and . Duruk et al. investigated the well posedness of the Cauchy problem and showed that under certain conditions the Cauchy problem is globally well posed [19]. Nevertheless, several types of the improved Boussinesq equation were investigated by many researchers and found exact solutions by using exp-function method [20], modified extended tanh-function method [21], sine-cosine method [22], improved G’/G-expansion method [22], the standard tanh and the extended tanh method [23], and so forth.

The tanh-coth is a powerful and reliable technique for finding exact travelling wave solutions for nonlinear equations. This method has been used extensively, and it was subjected by some modifications using the Riccati equation. The main features of the tanh-coth method will be outlined in the subsequent section, and this method will be applied to the the Benney-Luke and the Higher-order improved Boussinesq equations. The main purpose of this work is to obtain travelling wave solutions of the above-mentioned equations and to show that the tanh-coth method can be easily applied to Sobolev type equations. Throughout the work, Maple is used to deal with the tedious algebraic operations.

2. Outline of the Tanh-Coth Method

Wazwaz has summarized the tanh method in the following manner.(i)First consider a general form of nonlinear equation (ii)To find the traveling wave solution of (2.1), the wave variable is introduced, so that Based on this one may use the following changes: and so on for other derivatives. Using (2.3) changes the PDE (2.1) to an ODE as follows: (iii)If all terms of the resulting ODE contain derivatives in , then by integrating this equation and by considering the constant of integration to be zero, one obtains a simplified ODE.(iv)A new independent variable is introduced that leads to the change of derivatives: where other derivatives can be derived in a similar manner.(v)The ansatz of the form is introduced where is a positive integer, in most cases, that will be determined. If is not an integer, then a transformation formula is used to overcome this difficulty. Substituting (2.6) and (2.7) into the ODE, (2.4) yields an equation in powers of .(vi)To determine the parameter , the linear terms of highest order in the resulting equation with the highest order nonlinear terms are balanced. With determined, one collects the all coefficients of powers of in the resulting equation where these coefficients have to vanish. This will give a system of algebraic equations involving the and , (), , and . Having determined these parameters, knowing that is a positive integer in most cases, and using (2.7) one obtains an analytic solution in a closed form.

3. The Benney-Luke Equation

The Benney-Luke equation can be written as where and are positive numbers such that ( is named the Bond number). In order to solve (3.1) by the tanh-coth method, we use the wave transformation with wave variable ; (3.1) takes on the form of an ordinary differential equation as follows: Balancing the order of with the order of in (3.2) we find . Using the assumptions of the tanh-coth method (2.5)–(2.7) gives the solution in the form Substituting (3.3) into (3.2), we obtain a system of algebraic equations for , , , and in the following form: From the output of the Maple packages we find three sets of solutions: where is left as a free parameter. The travelling wave solutions are as follows:

4. The Higher-Order Improved Boussinesq Equation

We consider the Higher-order improved Boussinesq equation as follows: where and are arbitrary non zero real constants.

Using the wave transformation with wave variable then by integrating this equation and considering the constant of integration to be zero, we obtain the ODE as follows: Balancing the first term with the last term in (4.2) we find . Using the assumptions of the tanh-coth method (2.5)–(2.7) gives the solution in the form Substituting (4.3) into (4.2), we obtain a system of algebraic equations for , and in the following form: Using Maple gives six sets of solutions: The travelling wave solutions are as follows:

References

  1. R. W. Carroll and R. E. Showalter, Singular and Degenerate Cauchy Problems, vol. 12 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1976. View at Zentralblatt MATH · View at MathSciNet
  2. S. L. Sobolev, “Some new problems in mathematical physics,” Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, vol. 18, pp. 3–50, 1954.
  3. S. L. Sobolev, “On a new problem of mathematical physics,” Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, vol. 18, pp. 3–50, 1954. View at Zentralblatt MATH · View at MathSciNet
  4. S. A. Gabov, New Problems of the Mathematical Theory of Waves, Fizmatlit, Moscow, Russia, 1998.
  5. M. O. Korpusov, Yu. D. Pletner, and A. G. Sveshnikov, “Unsteady waves in media with anisotropic dispersion,” Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, vol. 39, no. 6, pp. 1006–1022, 1999. View at MathSciNet
  6. M. O. Korpusov and A. G. Sveshnikov, “Three-dimensional nonlinear evolution equations of pseudoparabolic type in problems of mathematical physics,” Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, vol. 43, no. 12, pp. 1835–1869, 2003. View at Zentralblatt MATH · View at MathSciNet
  7. P. I. Naumkin and I. A. Shishmarëv, Nonlinear Nonlocal Equations in the Theory of Waves, vol. 133 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1994. View at MathSciNet
  8. E. I. Kaĭkina, P. I. Naumkin, and I. A. Shishmarëv, “The Cauchy problem for a Sobolev-type equation with a power nonlinearity,” Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, vol. 69, no. 1, pp. 61–114, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. J. R. Quintero and J. C. Muñoz Grajales, “Instability of solitary waves for a generalized Benney-Luke equation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 68, no. 10, pp. 3009–3033, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. R. L. Pego and J. R. Quintero, “Two-dimensional solitary waves for a Benney-Luke equation,” Physica D, vol. 132, no. 4, pp. 476–496, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. D. J. Benney and J. C. Luke, “On the interactions of permanent waves of finite amplitude,” Journal of Mathematical Physics, vol. 43, pp. 309–313, 1964. View at Zentralblatt MATH · View at MathSciNet
  12. J. R. Quintero, “Nonlinear stability of solitary waves for a 2-D Benney-Luke equation,” Discrete and Continuous Dynamical Systems. Series A, vol. 13, no. 1, pp. 203–218, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  13. A. González N., “The Cauchy problem for Benney-Luke and generalized Benney-Luke equations,” Differential and Integral Equations, vol. 20, no. 12, pp. 1341–1362, 2007. View at Zentralblatt MATH · View at MathSciNet
  14. J. R. Quintero, “A remark on the Cauchy problem for the generalized Benney-Luke equation,” Differential and Integral Equations, vol. 21, no. 9-10, pp. 859–890, 2008. View at Zentralblatt MATH · View at MathSciNet
  15. S. Wang, G. Xu, and G. Chen, “Cauchy problem for the generalized Benney-Luke equation,” Journal of Mathematical Physics, vol. 48, no. 7, Article ID 073521, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. J. R. Quintero, “Existence and analyticity of lump solutions for generalized Benney-Luke equations,” Revista Colombiana de Matemáticas, vol. 36, no. 2, pp. 71–95, 2002. View at Zentralblatt MATH · View at MathSciNet
  17. L. Ji-bin, “Exact traveling wave solutions to 2D-generalized Benney-Luke equation,” Applied Mathematics and Mechanics, vol. 29, no. 11, pp. 1391–1398, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  18. G. Schneider and C. E. Wayne, “Kawahara dynamics in dispersive media,” Physica D, vol. 152-153, pp. 384–394, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. N. Duruk, A. Erkip, and H. A. Erbay, “A higher-order Boussinesq equation in locally non-linear theory of one-dimensional non-local elasticity,” IMA Journal of Applied Mathematics, vol. 74, no. 1, pp. 97–106, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. M. A. Abdou, A. A. Soliman, and S. T. El-Basyony, “New application of Exp-function method for improved Boussinesq equation,” Physics Letters, Section A, vol. 369, no. 5-6, pp. 469–475, 2007. View at Publisher · View at Google Scholar · View at Scopus
  21. S. A. Elwakil, S. K. El-Labany, M. A. Zahran, and R. Sabry, “Modified extended tanh-function method and its applications to nonlinear equations,” Applied Mathematics and Computation, vol. 161, no. 2, pp. 403–412, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. A. M. Wazwaz, “Nonlinear variants of the improved Boussinesq equation with compact and noncompact structures,” Computers & Mathematics with Applications, vol. 49, no. 4, pp. 565–574, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  23. J. Biazar and Z. Ayati, “Improved G'/G-expansion method and comparing with tanh-coth method,” Applications and Applied Mathematics, vol. 6, no. 11, pp. 1981–1991, 2011. View at MathSciNet