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

TL;DR

This paper studies how nonlinear dispersive waves with non-phase invariant Hamiltonians develop phase correlations, anomalous correlators, and negative frequencies, with analytical predictions confirmed by numerical simulations.

Contribution

It introduces a model showing the emergence of anomalous correlators and negative frequencies in nonlinear waves, with analytical results validated numerically.

Findings

Anomalous correlators develop on a timescale of O(1/ε).

Phase correlations between positive and negative wavenumbers emerge.

Numerical simulations confirm theoretical predictions.

Abstract

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.