Stability of stationary solutions in Acoustic wave turbulence

TL;DR

This work investigates the stability of stationary solutions to the Wave Kinetic Equation for acoustic waves, addressing both equilibrium Rayleigh-Jeans and non-equilibrium Kolmogorov-Zakharov spectra in 2D and 3D. By combining theoretical Carleman/Mellin analyses with numerical simulations (via the WavKinS framework on a logarithmic grid), the authors show RJ states are globally stable to isotropic perturbations, while KZ states are also stable but exhibit dimension-dependent perturbation dynamics, including velocity-mediated propagation and exponential damping. The study leverages isotropic reduction to a Carleman equation, uses the Mellin transform to characterize stability through the Mellin function, and applies Wiener-Hopf methods to construct solutions, revealing a rich structure in how perturbations decay or propagate. These results deepen understanding of energy fluxed wave turbulence in acoustic systems and outline limitations regarding anisotropic perturbations and higher-order effects, with implications for related wave-tuctuation models.

Abstract

We study the stability of steady-state solutions of the Wave-Kinetic Equations for acoustic waves. Combining theoretical analysis and numerical simulations, we characterise the time evolution of small isotropic perturbations for both 2D and 3D equilibrium Rayleigh-Jeans and non-equilibrium Kolmogorov-Zakharov solutions. In particular, we show that the stability of these solutions is ensured by different mechanisms.

Figures (11)

• Figure 1: KZ spectrum for acoustic waves. The solid black lines correspond to numerical simulations while red dashed lines represent the theoretical spectra $n_k = B k^{-\mu}$. (a) 2D case, $B = \dfrac{4}{3\pi^2} \sqrt{\dfrac{2aP}{c_s}}$griffin2022energy, $\mu = 3$. The inset present the initial spectral density of a small isotropic perturbation $A_k(0)$. (b) 3D case, $B = \dfrac{ \sqrt{6P}}{3\pi c_s \sqrt{32\pi(\pi + 4\ln 2 - 1)}}$zhu2024turbulence, $\mu = 9/2$. Where $P$ corresponds to the energy flux.
• Figure 2: Heatmaps of the perturbation for RJ solutions. (a) 2D case with $k_p/k_{\max} = 10^{-2}$. (b) 3D case with $k_p/k_{\max} = 10^{-2}$. (c) 2D case with $k_p/k_{\max} = 0.7$. (d) Same data as (c) shown with a saturated linear color scale. The top row illustrates similar behavior for both 2D and 3D simulations, characterized by a slow decay of the perturbation peak. The bottom row highlights that, for larger values of $k_p/k_{\max}$, the decay is significantly faster and associated with a clear broadening of the perturbation.
• Figure 3: Time evolution of the amplitude of a narrow Gaussian perturbation, centered in $k_p$, for the Rayleigh-Jeans solution. Colored dashed lines correspond to theoretical predictions $A(t) = A(0)e^{-\alpha(k_p) t}$, where $\alpha$ is given in Eq. \ref{['eq:alpha']}. $\color{red}{{}}$$k_p / k_{\max} = 10^{-2}$, $\color{purple}{{}}$$k_p /k_{\max} = 0.7$. (a) 2D case. (b) 3D case.
• Figure 4: Heatmap of the 2D isotropic perturbation $A(x,t)$; $x = \log{k/k_0}$. $x = V_{0} t$, where $V_{0} = 2\pi$ corresponds to the least damped velocity i.e. $\Gamma'(V_{0}) = 0$. (a) Numerical simulations using WavKinS krstulovic2025wavkins. (b) Numerical calculation of the integral allowing for arbitrarily large $x$, therefore allowing the observation of the combined limits $t \rightarrow +\infty \, ; \, x \rightarrow +\infty$. The numerical integration confirms the observed results of panel a and theoretical predictions of Eq. \ref{['eq:Asympt']} (dashed line).
• Figure 5: (a) Real part of the derivative of the Mellin function for 2D acoustics. (b) Real part of the second derivative of the Mellin function for 2D acoustics. Both figures share the same legend: $\color{red}{}$ Critical point $s_0^*\approx-0.11$, defined by $\mathrm{Re}(\mathop{\mathrm{\mathcal{W}}}\nolimits"(s_0^*)) = 0$, corresponding to the velocity $V_0 = \mathrm{Re}(\mathop{\mathrm{\mathcal{W}}}\nolimits'(s_0^*)) = 2\pi$. Negative real saddle points for $V \geq V_0$. $\color{blue}{}$ Complex saddle points for $V < V_0$.