A five-mode bifurcation analysis of a Kuramoto-Sivashinsky equation with dispersion

In several problems where a long wavelength oscillatory instability is found, the nonlinear evolution of the perturbations near criticality is governed by the dispersion modified Kuramoto-Sivashinsky equation (KS-KdV). It appears in problems of fluid flow along an inclined plane [ l-31, convection in fluids with a free surface [4], drift waves in plasmas [ 51, vertically falling liquid films in the presence of interfacial viscosities [ 61, etc. While in extended systems in the nondispersive case 6 = 0, i.e., in the KS equation, disordered behavior predominates, numerical studies have shown that for large dispersion the system evolves into rows of solitary like pulses of equal amplitude which travel as a whole [ 7-91. The number of pulses that appear depends on the initial conditions, the horizontal extension of the domain and on dispersion. Fourier analysis of the solutions obtained from time integrations of the partial differential equation show that the final state is composed of few modes and that stable modes decay rapidly [8]. An equilibrium between two modes, k and 2k, gives approximate agreement with some numerical results [ 5 1. A threemode system shows the secondary bifurcation from a steady state to rotating waves in the KS equation and the imperfect bifurcation when dispersion is included. Asymptotic results for large dispersion in the three-mode equilibria show that the solution evolves into a localized travelling pulse of amplitude which