Localized
Solutions in Parametrically Driven Pattern Formation
Dieter Armbruster
Arizona State University
The Mathieu partial differential equation is analyzed as a
prototypical model for pattern formation due to parametric resonance. After
averaging and scaling it is shown to be a perturbed Nonlinear
Schr\"{o}dinger Equation. Adiabatic perturbation theory for solitons is
applied to determine which solitons of the NLS survive the perturbation due to
damping and parametric forcing. Numerical simulations compare the perturbation
results to the dynamics of the Mathieu PDE. Stable and weakly unstable soliton
solutions are identified. They are shown to be closely related to oscillons
found in parametrically driven sand experiments.