Approximate solution of the Kuramoto-Shivashinsky equation on an unbounded domain

Abstract

The main goal of this paper is to approximate the Kuramoto-Shivashinsky (K-S for short) equation on an unbounded domain near a change of bifurcation, where a band of dominant pattern is changing stability. This leads to a slow modulation of the dominant pattern. Here we consider PDEs with quadratic nonlinearities and derive rigorously the modulation equation, which is called the Ginzburg-Landau (G-L for short) equation, for the amplitudes of the dominating modes.

Keywords

Multi-scale analysis / Modulation equation / Kuramoto-Shivashinsky equation / Ginzburg-Landau equation

References

[1] Adams R. A.. Sobolev spaces, 1975, New York: Academic Press

[2] Adams D. R., Hedberg L. I.. Function Spaces and Potential Theory. Springer-Verlag, 1996

[3] Adomian G.. Solving Frontier Problems of Physics: The Decomposition Method, 1994, Boston and London: Kluwer Academic Publishers

[4] Alizadeh-Pahlavan A., Aliakbar V., Vakili-Farahani F., Sadeghy K.. MHD flows of UCM fluids above porous stretching sheets using two-auxiliary-parameter homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul., 2009, 14: 47388

[5] Awrejcewicz J., Andrianov I. V., Manevitch L. I.. Asymptotic Approaches in Nonlinear Dynamics, 1998, Berlin: Springer-Verlag

[6] Baldwin D., Goktas O., Hereman W. Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs. J. Symbolic Comput., 2004, 37: 669-705

[7] Benney D. J.. Long waves on liquid films. J. Math. Phys., 1966, 45: 150-155

[8] Collet P., Eckmann J. P.. The time dependent amplitude equation for the Swift-Hohenberg problem. Comm. Math. Physics., 1990, 132: 139-153

[9] He J. H.. Homotopy perturbation technique. Comput. Meth. Appl. Mech. Eng., 1999, 178: 25762

[10] Hooper A. P., Grimshaw R.. Nonlinear instability at the interface between two fluids. Phys. Fluids, 1985, 28: 37-45

[11] Kuramoto Y.. Diffusion-induced chaos in reaction systems. Suppl. Prog. Theor. Phys, 1978, 64: 346-367

[12] Kuramoto Y., Tsuzuki T.. On the formation of dissipative structures in reaction diffusion systems. Prog. Theor. Phys., 1975, 54: 687-699

[13] Kuramoto Y., Tsuzuki T.. Persistent propogation of concentration waves in dissipative media far from thermal equilibrium. Prog. Theor. Phys., 1976, 55: 356-369

[14] Liao S. J.. A kind of approximate solution technique which does not depend upon small parameters (II): An application in fluid mechanics. Int. J. Nonlinear Mech., 1997, 32(8): 1522

[15] Ma W. X., Fuchssteiner B.. Integrable theory of the perturbation equations, Chaos. Solitons & Fractals, 1996, 7(8): 1227-1250

[16] Mielke A., Schneider G.. Attractors for modulation equations on unbounded domains-existence and comparison. Nonlinearity, 1995, 8: 743-768

[17] Mielke A., Schneider G., Ziegra A.. Comparison of inertial manifolds and application to modulated systems. Math. Nachr., 2000, 214: 53-69

[18] Runst T., Sickel W.. Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, Walter de Gruyter, 1996, New York: Berlin

[19] Sajid M., Hayat T.. Comparison of HAM and HPM methods for nonlinear heat conduction and convection equations. Nonlinear Anal: Real World Appl., 2008, 9: 2296301

[20] Schneider G.. A new estamite for the Ginzburg-Landau approximation on the real axis. J. Nonlinear Science, 1994, 4: 23-34

[21] Schneider G.. The validity of generalized Ginzburg-Landau equations. Math. Methods Appl. Sci., 1996, 19(9): 717-736

[22] Sivashinsky G. I.. Nonlinear analysis of hydrodynamic instability in laminar flames I: Derivation of basic equations. Acta Astronau, 1977, 4: 1177-1206

[23] Sivashinsky G. I.. Instabilities, pattern formation, and turbulence in flames. Annu. Rev. Fluid Mech., 1983, 15: 179-199

[24] Shlang T., Sivashinsky G. I.. Irregular flow of a liquid film down a vertical column. J. Phys., 1982, 43: 459-466

[25] Van Gorder R. A., Vajravelu R. A.. Analytic and numerical solutions to the Lane-Emden equation. Phys. Lett. A, 2008, 372: 60605

[26] Von Dyke M.. Perturbation Methods in Fluid Mechanics, The Parabolic Press, 1975, California: Stanford

[27] Yabushita K., Yamashita M., Tsuboi K.. An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method. J. Phys. A, 2007, 40: 840316

[28] Zhu S. P.. A closed-form analytical solution for the valuation of convertible bonds with constant dividend yield. ANZIAM J., 2006, 47: 47794