Secondary Instabilities and Spatiotemporal Chaos in Parametric Surface Waves

Abstract

A 2D model is introduced to study the onset of parametric surface waves, their secondary instabilities, and the transition to spatiotemporal chaos. We obtain the stability boundary of a periodic standing wave above onset against Eckhaus, zigzag, and transverse amplitude modulations (TAM), as a function of the control parameter $\ensuremath{\varepsilon}$ and the wavelength of the pattern. The Eckhaus and TAM boundaries cross at a finite value of $\ensuremath{\varepsilon}$, thus explaining the finite threshold for the TAM observed experimentally. At larger values of $\ensuremath{\varepsilon}$, a numerical solution reveals a transition to spatiotemporal chaotic states mediated by the TAM instability.