Stability and instability conditions for stationary acoustic WKE solutions

Determine the propagation dynamics of small perturbations and rigorously characterize the conditions under which instability may arise for stationary Rayleigh–Jeans equilibrium and Kolmogorov–Zakharov non‑equilibrium spectra of the acoustic Wave Kinetic Equation with three‑wave interactions, clarifying the mechanisms and thresholds responsible for stability or instability across two and three spatial dimensions.

Background

The paper studies the stability of steady-state solutions of the Wave Kinetic Equation (WKE) for acoustic waves, focusing on Rayleigh–Jeans (RJ) and Kolmogorov–Zakharov (KZ) spectra. Although these spectra are widely observed in experiments and simulations, a rigorous understanding of their stability properties—especially the propagation and fate of small perturbations and the precise conditions that trigger instability—has been limited.

The authors use analytical tools (including Mellin transforms and Carleman-type equations when applicable) and numerical simulations to analyze isotropic perturbations in 2D and 3D acoustic systems. This open question frames the broader motivation for their investigation and highlights the need for precise criteria delineating stability and potential instability of stationary solutions within the acoustic WKE framework.