Role of additional unstable modes in the secondary instability of capillary waves

Ascertain whether and how the presence of additional unstable modes accounts for the observed secondary instability that transforms initially monochromatic capillary waves into superpositions of multiple modes in the conserved active emulsion model, and characterize the mechanism responsible for this long-term behavior.

Background

After initial formation, finite-amplitude capillary waves often undergo a secondary instability, leading either to a stationary superposition of modes or to a time-dependent, quasi-chaotic multimode pattern.

The linear theory predicts bands of unstable modes, but simulations show that linearly stable modes can also gain weight in the long-term dynamics, suggesting an unresolved nonlinear mechanism possibly involving additional modes.