- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Annual Issues
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 890574, 10 pages
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.
- 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.
- S. L. Sobolev, “Some new problems in mathematical physics,” Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, vol. 18, pp. 3–50, 1954.
- S. L. Sobolev, “On a new problem of mathematical physics,” Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, vol. 18, pp. 3–50, 1954.
- S. A. Gabov, New Problems of the Mathematical Theory of Waves, Fizmatlit, Moscow, Russia, 1998.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- G. Schneider and C. E. Wayne, “Kawahara dynamics in dispersive media,” Physica D, vol. 152-153, pp. 384–394, 2001.
- 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.
- 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.
- 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.
- 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.
- J. Biazar and Z. Ayati, “Improved -expansion method and comparing with tanh-coth method,” Applications and Applied Mathematics, vol. 6, no. 11, pp. 1981–1991, 2011.