Coherent structures theory for the generalized Kuramoto-Sivashinsky equation

Abstract

We examine coherent structures interaction and formation of bound states in active–dispersive–dissipative nonlinear media. A prototype for such media is a simple weakly nonlinear model, the generalized Kuramoto-Sivashinsky (gKS) equation, that retains the fundamental mechanisms of any nonlinear process involving wave evolution, namely, a dominant nonlinearity, instability, stability and dispersion. We develop a weak interaction theory for the solitary pulses of the gKS equation by representing the solution as a superposition of the pulses and an overlap function. We derive a linearized equation for the overlap function in the vicinity of each pulse and project the dynamics of this function onto the discrete part of the spectrum of the linearized interaction operator. This leads to a coupled system of ordinary differential equations describing the evolution of the locations of the pulses. By analyzing this system, we prove a criterion for the existence of a countable infinite or finite number of bound states, depending on the strength of the dispersive term in the equation. The theoretical findings are corroborated by computations of the full equation.