Anomalous correlators, negative frequencies and non-phase-invariant Hamiltonians in random waves

We investigate a generic non-phase invariant Hamiltonian model that governs the dynamics of nonlinear dispersive waves. We give evidence that initial data characterized by random phases naturally evolve into phase correlations between positive and negative wavenumbers, leading to the emergence of non-zero anomalous correlators and negative frequencies. Using analytical techniques, we show that anomalous correlators develop on a timescale of $O(1/\epsilon)$, earlier than the kinetic timescale. Our theoretical predictions are validated through direct numerical simulations of the deterministic system.