Next-to-leading-order anisotropic stability analysis in acoustic WKE

Develop a rigorous stability framework for next‑to‑leading‑order anisotropic perturbations of the Rayleigh–Jeans and Kolmogorov–Zakharov spectra in the acoustic Wave Kinetic Equation, overcoming the loss of homogeneity caused by dispersive regularization that currently prevents derivation of a Carleman equation and application of the Balk–Zakharov method.

Background

For acoustic WKE, regularization via a small dispersive term is necessary to render collision integrals well-posed, particularly in 2D. While the Balk–Zakharov (BZ) approach provides a general stability framework, it relies on kernel homogeneity to derive Carleman equations. In the acoustic case, this homogeneity is broken beyond leading order due to regularization, preventing direct application of BZ theory to anisotropic perturbations at next-to-leading order.

The authors explicitly note that the present work focuses on leading-order isotropic perturbations and that extending the analysis to next-to-leading-order anisotropic perturbations is currently not feasible without new methods that handle the broken homogeneity. This identifies a concrete methodological gap and motivates future advances to treat anisotropic stability beyond leading order in acoustic wave turbulence.