Localized solutions in parametrically driven pattern formation.

The Mathieu partial differential equation (PDE) 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 Schrödinger equation (NLS). 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.