Stable Modulating Multipulse Solutions for Dissipative Systems with Resonant Spatially Periodic Forcing

Summary.

We show the existence and stability of modulating multipulse solutions for a class of bifurcation problems given by a dispersive Swift-Hohenberg type of equation with a spatially periodic forcing. Equations of this type arise as model problems for pattern formation over unbounded weakly oscillating domains and, more specifically, in laser optics. As associated modulation equation, one obtains a nonsymmetric Ginzburg-Landau equation which possesses exponentially stable stationary n—pulse solutions. The modulating multipulse solutions of the original equation then consist of a traveling pulselike envelope modulating a spatially oscillating wave train. They are constructed by means of spatial dynamics and center manifold theory. In order to show their stability, we use Floquet theory and combine the validity of the modulation equation with the exponential stability of the n—pulses in the modulation equation. The analysis is supplemented by a few numerical computations.

In addition, we also show, in a different parameter regime, the existence of exponentially stable stationary periodic solutions for the class of systems under consideration.