 
Summary: Localized solutions in parametrically driven pattern formation
TaeChang Jo and Dieter Armbruster*
Department of Mathematics, Arizona State University, Tempe, Arizona 852871804, USA
Received 28 October 2002; published 16 July 2003
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 Schro¨dinger
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 perturba
tion 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.
DOI: 10.1103/PhysRevE.68.016213 PACS number s : 89.75. k
I. INTRODUCTION
The standard model for the instability of a harmonic os
cillator due to parametric forcing parametric resonance is
the Mathieu equation 1 which can be written as
Att
2
cos qt A 0. 1
Here, is the eigenfrequency of the oscillator, is a small
forcing amplitude, and q is the forcing frequency. The trivial
