Stable modulating multipulse solutions for dissipative systems with resonant spatially periodic forcing

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 e quation, one obtains a nonsymmetric Ginzburg-Landau equation which possesse s exponentially stable stationary n-pulse solutions. The modulating multipu lse solutions of the original equation then consist of a traveling pulselik e envelope modulating a spatially oscillating wave train. They are construc ted by means of spatial dynamics and center manifold theory. In order to sh ow their stability, we use Floquet theory and combine the validity of the m odulation equation with the exponential stability of the n-pulses in the mo dulation equation. The analysis is supplemented by a few numerical computat ions. In addition, we also show, in a different parameter regime, the existence o f exponentially stable stationary periodic solutions for the class of syste ms under consideration.