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

TL;DR

This work shows that the spatio-temporal spectrum of weakly nonlinear dispersive waves contains pronounced features that cannot be explained by time resonances alone. By introducing space resonances—energy exchange between wave packets sharing a group velocity—the authors derive a leading-order spatio-temporal spectrum and nonlinear frequency shifts, illustrating how these interactions create branches such as $\sigma=2\omega(k/2)$ and $\sigma=\omega'(k_f)k$, and even negative frequencies when gauge-breaking terms are present. They validate the theory with Acqua Alta stereoscopic measurements and high-fidelity water-wave simulations, and extend the approach to simplified models to isolate the space-resonant contributions. Applying the framework to the full water-wave problem, they show that three- and four-wave interactions jointly generate higher harmonics and negative-frequency signatures, providing a cohesive explanation that complements the traditional wave-kinetic, time-resonance picture and enhances predictive capability for laboratory and oceanic wave fields.

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.

Figures (4)

• Figure 1: Spatio-temporal spectrum of a linear combination of surface elevation and velocity potential on the surface, $|a(k,\sigma)|^2$, from the numerical simulation of water wave equations. Quantities in the figure are non-dimensional, corresponding to rescaled quantities in the simulation. Five branches discussed in the text are marked in the figure, including the main branch (fitted by $\sigma=\sqrt{g|k|}$ for negative $k$), branch A ($\sigma=\sqrt{g|k|}$ for positive $k$), branch B ($\sigma=\sqrt{2g|k|}$ for negative $k$), branch C (approximately by $\sigma = c_g k$ near the origin) and branch D ($\sigma=-\sqrt{g|k|}$ for positive $k$).
• Figure 2: Spatio-temporal spectrum of sea surface elevation measured at the Acqua Alta oceanographic tower, situated 15 km offshore from Venice. The red dashed lines correspond, starting from the top one, to $\sigma = \sqrt{2g|k|}, \sigma = \sqrt{g|k|}, \sigma =- \sqrt{g|k|}, \sigma =- \sqrt{2g|k|}$
• Figure 3: The spatio-temporal spectra obtained from numerical simulations of the equation in \ref{['eq:3waves']} with dispersion relation $\omega(k)=\sqrt{g|k|}$ (left plot) and $\omega(k)=|k|^2$ (right plot). Both plots show different excitations: the solid black curves in each plot corresponds to the dispersion relation for the free waves, while blue dashed lines are attributed to bound modes. In the left plot, the equation for the bound modes are $\sigma(k)=\pm \sqrt{2 g |k|}$ and $\sigma(k)=c_g k$, where $c_g$, is the group velocity computed at the peak of the spectrum, $k\simeq 1$. Similarly, in the right plot, the equation for the bound modes are $\sigma(k)=|k|^2/2$ and $\sigma(k)=c_g k$, where $c_g$, is the group velocity computed at the peak of the spectrum, $k\simeq 1$.
• Figure 4: (a) Naive approximation of the theoretical spatio-temporal spectrum $S_k$ from the simplified model, showing all triadic branches, assuming the coefficients $A^{(i,j)}=1$, $\omega_k = \sqrt{g |k|}$ with $g=1$ and resolution $L = 40$. In other words, we plot the support set of the spatio-temporal spectrum. Specifically, above the $\sigma=\omega(k)$ curve, $\sum_{k_2+k_3=k} \delta(\sigma -\omega(k_2)-\omega(k_3))$ is plotted: for each $k$ we plot $\sum _{k_2}\mathcal{\chi}_{\sigma=\omega(k-k_2)-\omega(k_2)}$ where $\mathcal{\chi}$ is the indicator function and $k_2$ runs over the wavenumbers on the lattice. The dots corresponds to all values of $\sigma$ where the indicator function does not vanish. The curves of different colors correspond to the sets of support of the indicator functions in the sum in the continuum limit $L\rightarrow\infty$. This is equivalent to a naive approximation of the normal form transformation. (b) The space resonances of the leading order expansion and the linear dispersion relation.

Theorems & Definitions (2)

• Definition 3.1: Space Resonance
• Remark 3.2