Oscillons: localized patterns in a periodically forced system

Spatially localized, time-periodic structures, known as oscillons, are common in patternforming
systems, appearing in fluid mechanics, chemical reactions, optics and granular
media. This thesis examines the existence of oscillatory localized states in a PDE model
with single frequency time dependent forcing, introduced in [70] as phenomenological
model of the Faraday wave experiment. Firstly in the case where the prefered
wavenumber at onset is zero, we reduce the PDE model to the forced complex Ginzburg–
Landau equation in the limit of weak forcing and weak damping. This allows us to use
the known localized solutions found in [15]. We reduce the forced complex Ginzburg–
Landau equation to the Allen–Cahn equation near onset, obtaining an asymptotically
exact expression for localized solutions. In the strong forcing case, we get the Allen–Cahn
equation directly. Throughout, we use continuation techniques to compute numerical
solutions of the PDE model and the reduced amplitude equation. We do quantitative
comparison of localized solutions and bifurcation diagrams between the PDE model, the
forced complex Ginzburg–Landau equation, and the Allen–Cahn equation. The second
aspect in this work concerns the investigation of the existence of localized oscillons
that arise with non-zero preferred wavenumber. In the limit of weak damping, weak
detuning, weak forcing, small group velocity, and small amplitude, asymptotic reduction
of the model PDE to the coupled forced complex Ginzburg–Landau equations is done.
In the further limit of being very close to onset, we reduce the coupled forced complex
Ginzburg–Landau equations to the real Ginzburg–Landau equation. We have qualitative
prediction of finding exact localized solutions from the real Ginzburg–Landau equation
limited by computational constraints of domain size. Finally, we examine the existence
of localized oscillons in the PDE model with cubic–quintic nonlinearity in the strong
damping, strong forcing and large amplitude case. We find two snaking branches in the
bistability region between stable periodic patterns and the stable trivial state in one spatial
dimension in a manner similar to systems without time dependent forcing. We present
numerical examples of localized oscillatory spots and rings in two spatial dimensions.