Coherent Structures in Convection and Parametrically Driven Surface Waves

A system maintained outside of thermodynamic equilibrium often develops complex spatial structures and, in some cases, persistent dynamics. The amplitude equation formalism has been widely used to study pattern formation in the weakly nonlinear regime (slightly beyond onset of instability from some stationary reference state.) Application of the formalism is briefly discussed for Rayleigh-B'enard convection, and at some length for parametrically driven surface waves (Faraday waves). I. INTRODUCTION To understand the emergence of coherent structures in a system that is driven outside of thermodynamic equilibrium, one must begin with a stability analysis of a reference stationary state that becomes marginally stable for some particular value of an experimentally set control parameter. Let /(x) be a particular state variable of a one-dimensional system, and " a dimensionless control parameter: at " = 0 the base state / = 0 loses stability against infinitesimal perturbations of the form,..