We examine weak interaction and formation of bound states of pulses for the generalized Kuramoto–Sivashinsky (gKS) equation, which is one of the simplest prototypes describing active media with energy supply, dissipation, dispersion, and nonlinearity. We derive a system of ordinary differential equations describing the leading-order dynamics of the pulses of the gKS equation and prove a criterion for the existence of a countable infinite or finite number of bound states. Our theory is corroborated by computations of the full equation.
Chang, H.-C. and Demekhin, E. A. 2002. Complex Wave Dynamics on Thin Films. Elsevier Scientific, Amsterdam, the Netherlands.
Cohen, B. I., Krommes, J. A., Tang, W. M., and Rosenbluth, M. N. 1976. Non-linear saturation of the dissipative trapped ion mode by mode coupling. Nucl. Fusion, 16, 971–992.
Duprat, C., Giorgiutti-Dauphiné, F., Tseluiko, D., Saprykin, S., and Kalliadasis, S. 2009. Liquid film coating a fiber as a model system for the formation of bound states in active dispersive–dissipative nonlinear media. Phys. Rev. Lett., 103, 234501.
doi:10.1103/PhysRevLett.103.234501
Ei, S.-I. 2002. The motion of weakly interacting pulses in reaction–diffusion systems. J. Dyn. Differ. Equ., 14, 85–137.
doi:10.1023/A:1012980128575
Ei, S.-I. and Ohta, T. 1994. Equation of motion for interacting pulses. Phys. Rev. E, 50, 4672–4678.
doi:10.1103/PhysRevE.50.4672
Elphick, C., Ierley, G. R., Regev, O., and Spiegel, E. A. 1991. Interacting localized structures with Galilean invariance. Phys. Rev. A, 44, 1110–1122.
doi:10.1103/PhysRevA.44.1110
Feng, B.-F., Malomed, B. A., and Kawahara, T. 2002. Stable periodic waves in coupled Kuramoto–Sivashinsky–Korteweg–de Vries equation. J. Phys. Soc. Jpn., 71, 2700–2707.
doi:10.1143/JPSJ.71.2700
Glendinning, P. and Sparrow, C. 1984. Local and global behaviour near homoclinic orbits. J. Stat. Phys., 35, 645–696.
doi:10.1007/BF01010828
Jurman, L. A. and McCready, M. J. 1989. Study of waves on thin liquid films sheared by turbulent gas flows. Phys. Fluids A, 1, 522–536.
doi:10.1063/1.857553
Kalliadasis, S., Demekhin, E. A., Ruyer-Quil, C., and Velarde, M. G. 2003. Thermocapillary instability and wave formation on a film flowing down a uniformly heated plane. J. Fluid Mech., 492, 303–338.
doi:10.1017/S0022112003005809
Kawahara, T. 1983. Formation of saturated solitons in a nonlinear dispersive system with instability and dissipation. Phys. Rev. Lett., 51, 381–383.
doi:10.1103/PhysRevLett.51.381
Kawahara, T. and Toh, S. 1988. Pulse interaction in an unstable dissipative-dispersive nonlinear system. Phys. Fluids, 11, 2103–2111.
doi:10.1063/1.866610
Oron, A. and Edwards, D. A. 1993. Stability of a falling liquid film in the presence of interfacial viscous stress. Phys. Fluids A, 5, 506–508.
doi:10.1063/1.858875
Pego, R. L. and Weinstein, M. I. 1992. Eigenvalues, and instabilities of solitary waves. Phil. Trans. R. Soc. Lond. A, 340, 47–94.
doi:10.1098/rsta.1992.0055
Promislow, K. 2002. A renormalization method for modulational stability of quasi-steady patterns in dispersive systems. SIAM J. Math. Anal., 33, 1455–1482.
doi:10.1137/S0036141000377547
Sandstede, B. 2002. Stability of travelling waves. In Handbook of Dynamical Systems II} (Fiedler, B., ed.), pp. 983–1055. North-Holland.
doi:10.1016/S1874-575X(02)80039-X
Trefethen, L. N. 2000. Spectral Methods in Matlab. Society for Industrial and Applied Mathematics, Philadelphia, PA.
