Secondary Instabilities and Spatiotemporal Chaos in Parametric Surface Waves

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 $\varepsilon$ and the wavelength of the pattern. The Eckhaus and TAM boundaries cross at a finite value of $\varepsilon$, thus explaining the finite threshold for the TAM observed experimentally. At larger values of $\varepsilon$, a numerical solution reveals a transition to spatiotemporal chaotic states mediated by the TAM instability.