Space-time resonances in the spatiotemporal spectrum of nonlinear dispersive waves

TL;DR

This paper introduces the concept of space resonances in nonlinear dispersive waves, explaining long-lived interactions observed in spatio-temporal spectra beyond traditional time resonance theory.

Contribution

It develops a theoretical framework for space resonances, supported by data, simulations, and analysis, expanding understanding of wave interactions in dispersive systems.

Findings

Identification of space resonances as a new mechanism in wave dynamics

Derivation of the spatio-temporal spectrum for water waves

Explanation of negative frequency support due to gauge-breaking terms

Abstract

In weakly nonlinear dispersive wave systems, long-time dynamics are typically governed by time resonances, where wave phases evolve coherently due to exact frequency matching. Recent advances in spatio-temporal spectrum measurements, however, reveal prominent features that go beyond the predictions of time resonance theory. In this work, we develop a theoretical framework to interpret these signatures by identifying and characterizing an alternative mechanism: space resonances. These arise when wave packets share the same group velocity and remain co-located, leading to long-lived interactions. We further show that gauge-breaking terms in the Hamiltonian give rise to space resonances supported on negative frequencies. By combining sea-surface elevation data, numerical simulations, and analytical theory, we derive the leading-order spatio-temporal spectrum for weakly interacting water waves, providing a unified explanation for its observed features.