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Localized solutions in parametrically driven pattern formation Tae-Chang Jo and Dieter Armbruster*
 

Summary: Localized solutions in parametrically driven pattern formation
Tae-Chang Jo and Dieter Armbruster*
Department of Mathematics, Arizona State University, Tempe, Arizona 85287-1804, 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 Schrošdinger
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

  

Source: Armbruster, Dieter - Department of Mathematics and Statistics, Arizona State University

 

Collections: Mathematics