Design of parametrically forced patterns and quasipatterns

TL;DR

This paper designs parametric forcing functions to generate superlattice patterns and quasipatterns in a PDE model, analyzing the limitations of three-wave interactions and providing examples of high-fold approximate quasipatterns.

Contribution

It introduces a method to design forcing functions for pattern selection in PDEs and explores the role of three-wave interactions in stabilizing quasipatterns.

Findings

Good agreement near pattern onset with theoretical predictions

Limited validity range due to damping constraints

Examples of 12-, 14-, and 20-fold approximate quasipatterns

Abstract

The Faraday wave experiment is a classic example of a system driven by parametric forcing, and it produces a wide range of complex patterns, including superlattice patterns and quasipatterns. Nonlinear three-wave interactions between driven and weakly damped modes play a key role in determining which patterns are favoured. We use this idea to design single and multi-frequency forcing functions that produce examples of superlattice patterns and quasipatterns in a new model PDE with parametric forcing. We make quantitative comparisons between the predicted patterns and the solutions of the PDE. Unexpectedly, the agreement is good only for parameter values very close to onset. The reason that the range of validity is limited is that the theory requires strong damping of all modes apart from the driven pattern-forming modes. This is in conflict with the requirement for weak damping if three-wave coupling is to influence pattern selection effectively. We distinguish the two different ways that three-wave interactions can be used to stabilise quasipatterns, and present examples of 12-, 14- and 20-fold approximate quasipatterns. We identify which computational domains provide the most accurate approximations to 12-fold quasipatterns, and systematically investigate the Fourier spectra of the most accurate approximations.