Traveling Wave Solutions of the RLW and Boussinesq Equations

Abstract

In this study, we use the generalized tanh function method for the traveling wave solutions of the generalized regularized long-wave (gRLW) equation and Boussinesq equation system

Keywords

• The generalized regularized long-wave equation,Boussinesq equation system,generalized tanh function method,traveling wave solutions

Traveling Wave Solutions of the RLW and Boussinesq Equations

Öz

References

1. Debtnath L. Nonlinear Partial Differential Equations for Scientist and Engineers. Birkhauser, Boston, MA, 1997.
2. Wazwaz A.M. Partial Differential Equations: Methods and Applications. Balkema, Rotterdam, 2002.
3. Hereman W., Banerjee P.P., Korpel A., Assanto G., Immerzeele A. van, Meerpoel A. Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method. J. Phys. A: Math. Gen. 19 (1986) p. 607-628.
4. Khater A.H., Helal M.A., El-Kalaawy O.H. Bäcklund transformations: exact solutions for the KdV and the Calogero-Degasperis-Fokas mKdV equations. Math. Meth. in the Appl. Sci. 21 (1998) p.719-731.
5. Khater A.H., Malfiet W., Kamel E.S. Travelling wave solutions of some classes of nonlinear evolution equations in (1+1) and higher dimensions. Math. Comput. Simul 64 (2004) p.247-258.
6. M. Inc. Constructing solitary pattern solutions of the nonlinear dispersive Zakharov–Kuznetsov equation. Chaos Solitons Fract. 39 (2009) p.109-119.
7. Khater A.H., Hassan M.M., Temsah R.S. Cnoidal wave solutions for a class of fifth-order KdV equations. Math. Comput. Simul. 70 (2005) p.221-226.
8. Ugurlu Y., Kaya D. Solutions the Cahn-Hilliard Equation. Comput. & Math. with Appl. 56 (2008) p.3038-3045.

9. Khater A.H., Callebaut D.K., Seadawy A.R. General soliton solutions of an n-dimensional complex Ginzberg-Landau equation. Phys. Scr. 62 (2000) p.353-357.
10. Khater A.H., Hassan M.M., Temsah R.S. Exact solutions with Jacobi elliptic functions of two nonlinear models for ion acoustic plasma wave. J. Phys. Soc. Japan 74 (2005) p1431-1435.
11. Inan I.E. Exact solutions for coupled KdV equation and KdV equations. Phys. Lett. A 371 (2007) p.90-95.
12. Oziş T., Yıldırım A. Reliable analysis for obtaining exact soliton solutions of nonlinear Schrödinger (NLS) equation. Chaos Solitons Fract. 38 (2008), p. 209-212.
13. M. Inc. An L-stable extended two-step method for the integration of ordinary differential equations. Appl. Math. Comput. 186 (2007) 1395- 140
14. M. Inc. An approximate solitary wave solution with compact support for the modified KdV equation. Appl. Math. Comput. 184 (2007) p.631-637.
15. Uğurlu Y., Kaya D. Exact and Numerical Solutions of Generalized Drinfeld-Sokolov Equations. Phys. Lett. A 372 (2008) p.2876-2873.
16. Fan E.G. Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A 277 (2000) p.212-218.
17. Chen H., Zhang H. New multiple soliton solutions to the general Burgers–Fisher equation and the Kuramoto–Sivashinsky equation. Chaos Solitons Fract. 19 (2004) p.71-76.
18. Hereman W., Korpel A., Banerjee P.P. Wave Motion 7 (1985) p.283-289.
19. Hereman W., Takaoka M. Solitary wave solutions of nonlinear evolution and wave equations using a direct method and MACSYMA. J. Phys. A: Math. Gen. 23 (1990) p.4805-4822.
20. Lan H., Wang K. Exact solutions for two nonlinear equations. J. Phys. A: Math. Gen. 23 (1990) p.3923-3928.
21. Lou S., Huang G., Ruan H. Exact solitary waves in a convecting fluid. J. Phys. A: Math. Gen. 24 (1991) L587-L590.
22. Malfliet W. Solitary wave solutions of nonlinear wave equations, Am. J. Phys. 60 (1992) p.650-654.
23. Parkes E. J., Duffy B. R. An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations. Comput. Phys. Commun. 98 (1996) p.288-300.
24. Fan E. Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A 277 (2000) p.212-218.
25. Elwakil S. A., El-labany S. K., Zahran M. A., Sabry R. Modified extended tanh-function method for solving nonlinear partial differential equations. Phys. Lett. A 299 (2002) p.179-188.
26. Zheng X., Chen Y., Zhang H. Generalized extended tanh-function method and its application to (1+1)-dimensional dispersive long wave equation. Phys. Lett. A 311 (2003) p.145-157.
27. Yomba E. Construction of new soliton-like solutions of the (2+1) dimensional dispersive long wave equation. Chaos Solitons Fract. 20 (2004) p.1135-1139.
28. Chen H., Zhang H. New multiple soliton solutions to the general Burgers-Fisher equation and the Kuramoto-Sivashinsky equation. Chaos Solitons Fract.19 (2004) p.71-76.
29. Chen H., Zhang H. New multiple soliton-like solutions to the generalized (2 + 1)-dimensional KP equation. Appl. Math. and Comput. 157 (2004) p.765-773.
30. Peregrine D.H. Calculations of the development of an undular bore. J. Fluid Mech. 25 (1996) p.321-330.
31. Peregrine D.H. Long waves on a beach,.J. Fluid Mech. 27 (1967) p.815-827.
32. Benjamin T.B., Bona J.L., Mahony J.J. Model equations for long waves in non-linear dispersive systems. Phil. Trans. of the Royal Soc. 272A (1972) p.47-78.
33. Bona J.L., Pritchard W.G., Scott L.R.. A comparison of solutions of two model equations for long waves, in: N.R. Lebovitz (Ed.), Fluid Dynamics in Astrophysics and Geophysics, Lectures in Appl. Math. (1983) p.235–267.
34. Bona J.L., Pritchard W.G., Scott L.R. An evaluation of a model equation for water waves. Phil. Trans. of the Royal Soc. 302 A (1981) 457- 5
35. Sachs R.L. On the integrable variant of the boussinesq system: Painlevé property, rational solutions, a related many-body system, and equivalence with the AKNS hierarchy. Physica D 30 (1988) p.1-27.