Abstract
In this paper, we consider a dissipative-dispersive nonlinear equation appliable to many physical phenomena. Using the geometric singular perturbation method based on the theory of dynamical systems, we investigate the existence of its traveling wave solutions with the dissipative terms having sufficiently small coefficients. The results show that the traveling waves exist on a two-dimensional slow manifold in a three-dimensional system of ordinary differential equations (ODEs). Then, we use the Melnikov method to establish the existence of a homoclinic orbit in this manifold corresponding to a solitary wave solution of the equation. Furthermore, we present some numerical computations to show the approximations of such wave orbits.
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References
Kliakhandler, I. L., Porubov, A. V., and Velarde, M. G. Localized finite-amplitude disturbances and selectionof solitary waves. Physical Review E 62, 4959–4962 (2000)
Lou, S., Huang, G., and Ruan, H. Exact solitary waves in a convecting fluid. Journal of Physics A 24, L587–L590 (1991)
Porubov, A. V. Exact travelling wave solutions of nonlinear evolution of surface waves in a convecting fluid. Journal of Physics A 26, L797–L800 (1993)
Velarde, M., G., Nekorkin, V. I., and Maksimov, A. G. Further results on the evolution of solitary waves and their bound states of a dissipative Korteweg-de Vries equation. International Journal of Bifurcation and Chaos 5, 831–839 (1995)
Fenichel, N. Geometric singular perturbation theory for ordinary differential equations. Journal of Differential Equations 31, 53–98(1979)
Jones, C. K. R. T. Geometric singular perturbation theory. Dynamical Systems (ed. Johnson, R.), Springer-Verlag, Berlin (1995)
Ruan, R. and Xiao, D. Stability of steady states and existence of travelling waves in a vector-disease model. Proceedings of the Royal Society of Edinburgh: Section A 134, 991–1011 (2004)
Ktrychko, Y. N., Bartuccelli, M. V., and Blyuss, K. B. Persistence of traveling wave solutions of a fourth order diffusion system. Journal of Computational and Applied Mathematics 176, 433–443 (2005)
Mansour, M. B. A. Existence of traveling wave solutions in a hyperbolic-elliptic system of equations. Communications in Mathematical Sciences 4, 731–739 (2006)
Guckenheimer, J. and Holmes, P. Nonlinear Oscillations, Dynamical System, and Bifurcation of Vector Fields, Springer Verlag, New York (1983)
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(Communicated by Li-qun CHEN)
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Mansour, M.B.A. Existence of traveling wave solutions for a nonlinear dissipative-dispersive equation. Appl. Math. Mech.-Engl. Ed. 30, 513–516 (2009). https://doi.org/10.1007/s10483-009-0411-6
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DOI: https://doi.org/10.1007/s10483-009-0411-6