Weak interaction of solitary pulses in active dispersive-dissipative nonlinear media

We develop an accurate weak-interaction theory for the pulses of the generalized Kuramoto-Sivashinsky (gKS) equation. This equation is the simplest prototype that retains the fundamental mechanisms of wave evolution in nonlinear media, namely, dominant nonlinearity, instability/energy supply, stability/energy dissipation and dispersion. The dynamics of the usual (dispersionless) KS equation is chaotic in both space and time. However, sufficiently strong dispersion regularizes the dynamics and the solutions evolve into arrays of interacting pulses, that can form bound states if dispersion does not exceed a certain threshold value. To obtain a theoretical insight into the interaction of the pulses, we represent a solution of the gKS equation as a superposition of pulses and an overlap function, and we carefully derive a coupled system of ordinary differential equations describing the evolution of the locations of the pulses by projecting the dynamics onto translational modes. This approach allows one to analyse bound states of the pulses and to derive a criterion on the existence of a countable infinite or finite number of bound states, depending on the strength of the dispersive term in the equation, which is in agreement with Shilnikov's criterion on the existence of subsidiary homoclinic orbits. We consider in detail two- and three-pulse bound states. We compare the interaction theory with computations of the full equation when the initial condition is a superposition of two or three pulses, and in all cases we find very good agreement.