Existence of Traveling Wave Solutions for a Nonlinear Dissipative-Dispersive Equation

M. B. A. Mansour

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2009 > 30(4): 479-483

M. B. A. Mansour. Existence of Traveling Wave Solutions for a Nonlinear Dissipative-Dispersive Equation[J]. Applied Mathematics and Mechanics, 2009, 30(4): 479-483.

Citation:

M. B. A. Mansour. Existence of Traveling Wave Solutions for a Nonlinear Dissipative-Dispersive Equation[J]. Applied Mathematics and Mechanics, 2009, 30(4): 479-483.

Citation:

M. B. A. Mansour. Existence of Traveling Wave Solutions for a Nonlinear Dissipative-Dispersive Equation[J]. Applied Mathematics and Mechanics, 2009, 30(4): 479-483.

PDF( 223 KB)

Existence of Traveling Wave Solutions for a Nonlinear Dissipative-Dispersive Equation

M. B. A. Mansour

Mathematics Department, Faculty of Science at Qena, South Valley University, Qena, Egypt

Received Date: 2008-06-09
Rev Recd Date: 2009-03-06
Publish Date: 2009-04-15

Abstract

Abstract

A dissipative-dispersive nonlinear equation which appears in many physical phenomena is considered.By using dynamical systems method,specifically the geometric singular perturbation method,the existence of traveling wave solutions of the equation when the dissipative terms have sufficiently small coefficients was investigated.It was shown that the traveling waves exist on a two-dimensional slow manifold in a three-dimensional system of ODEs.Then,by using the Melnikov method,the existence of a homoclinic orbit in this manifold,which corresponds to a solitary wave solution of the equation,was established.Furthermore,some numerical computations were presented to show approximations of such wave orbits.
- dissipative-dispersive equation,
- singular perturbations,
- traveling waves

FullText(HTML)

References(10)

References

[1]	Kliakhandler I L, Porubov A V, Velarde M G.Localized finite-amplitude disturbances and selection of solitary waves[J].Phys Rev E,2000,62：4959-4962. doi: 10.1103/PhysRevE.62.4959
[2]	Lou S Y, Huang G X, Ruan H Y. Exact solitary waves in a convecting fluid[J].J Phys A,1991,24(11): L587-L590.
[3]	Porubov A V. Exact travelling wave solutions of nonlinear evolution equation of surface waves in a convecting fluid[J].J Phys A,1993,26(17): L797-L800.
[4]	Velarde M G, Nekorkin V I, Maksimov A G. Further results on the evolution of solitary waves and their bound states of a dissipative Korteweg-de Vries equation[J].Internat J Bifurcation Chaos,1995,5(3): 831-839. doi: 10.1142/S0218127495000612
[5]	Fenichel N. Geometric singular perturbation theory for ordinary differential equations [J].J Differantial Equations,1979,31(1): 53-98. doi: 10.1016/0022-0396(79)90152-9
[6]	Jones C K R T. Geometric singular perturbation theory[A].In: Johnson R , Ed.Dynamical Systems[C].Berlin, Heidelberg: Springer-Verlag, 1995.
[7]	Ruan S G, Xiao D M. Stability of steady states and existence of travelling waves in a vector-disease model[J].Proc Roy Soc Edinburgh, Sect A,2004,134(5): 991-1011. doi: 10.1017/S0308210500003590
[8]	Ktrychko Y N, Bartuccelli M V, Blyuss K B. Persistence of traveling wave solutions of a fourth order diffusion system[J].J Comput Appl Math,2005,176(2): 433-443. doi: 10.1016/j.cam.2004.07.028
[9]	Mansour M B A. Existence of traveling wave solutions in a hyperbolic-elliptic system of equations[J].Comm Math Sci,2006,4:731-739.
[10]	Guckenheimer J, Holmes P.Nonlinear Oscillations, Dynamical System, and Bifurcation of Vector Fields[M]. New York: Springer-Verlag, 1983.