Kolmogorov–Zakharov Solutions in Wave Turbulence

• Kolmogorov–Zakharov solutions are power-law stationary spectral distributions that model constant-flux cascades via resonant nonlinear wave interactions.
• Analytical techniques such as Mellin transforms and winding number analysis establish their stability in both 2D and 3D turbulence regimes.
• These solutions apply to varied systems—from acoustic turbulence to optical fibers—where experimental realizability hinges on inertial ranges and dissipation control.

Kolmogorov–Zakharov (KZ) solutions are a distinguished class of stationary spectral distributions for wave kinetic equations, arising in the weak turbulence theory for nonlinear dispersive systems. These spectra embody non-equilibrium, steady flux states in which energy, mass, or other invariants are macroscopically transferred between scales via resonant nonlinear wave–wave interactions. KZ solutions are fundamental to understanding the statistical theory of turbulence in systems ranging from acoustic and capillary waves to plasmas, nonlinear optics, and elastic plates. Their mathematical characterization, stability, and physical realizability are central, active topics in contemporary turbulence theory.

1. Mathematical Formulation and Steady-State Solutions

The KZ spectra are power-law, stationary solutions of kinetic equations governing the spectral wave density $n_k$ or $f(\omega)$. For acoustic waves, the isotropic wave–kinetic equation (WKE) in $D$ spatial dimensions is typically formulated as:

$$\partial_t n_k = C_D \left[ \int_0^k [k_1 (k-k_1)]^{D-1}(n_1 n_2 - n_k(n_1 + n_2)) dk_1 - 2 \int_k^\infty [k_1 (k_1-k)]^{D-1} (n_k n_2 - n_1(n_k + n_2)) dk_1 \right]$$

where $C_D$ is a dimension-dependent constant, and the notation refers to wavevectors $k_1$, $k_2 = k - k_1$, $n_1 = n_{k_1}$, $n_2 = n_{k_2}$.

Steady-state solutions fall into two main classes:

• Rayleigh–Jeans (RJ) Equilibrium: $n_k^0 \propto [\omega(k)]^{-1}$, e.g., $n_k^0 \propto 1/(c_s k)$ for linear dispersion, representing thermal equilibrium with zero flux.
• Kolmogorov–Zakharov (KZ) Spectrum: Power-law non-equilibrium stationary solutions, $n_k^0 = B k^{-\mu}$, with $\mu$ determined by the resonance and dimension. In acoustic turbulence, $\mu=3$ in 2D, $\mu=9/2$ in 3D (Costa et al., 13 Aug 2025).

These KZ spectra encode constant-flux cascades: for example, the direct energy cascade (towards high $k$) or inverse mass cascade (towards small $k$) (Collot et al., 2022), the distinction being model and context dependent.

2. Linear Stability of KZ and RJ Spectra

To assess the stability of KZ solutions, one considers small isotropic perturbations $n_k = n_k^0(1 + A_k(t))$, $A_k \ll 1$, leading, upon linearization, to an evolution for $A_k$ governed by an integro-differential operator with convolutional and homogeneous structure.

Mellin Transform and Carleman Equation

By taking the Mellin transform,

$$\hat{A}(s, t) = \int_0^\infty A(k, t)\, k^{s-1}\,dk,$$

the linearized evolution recasts as a Carleman-type equation:

$$\partial_t \hat{A}(s+h, t) = \tau^{-1} \mathcal{W}(s) \hat{A}(s, t)$$

where $h$ is the kernel's homogeneity and $\mathcal{W}(s)$ is the Mellin (stability) function.

2D and 3D Acoustic Turbulence

• 2D ($h=0$): $$\mathcal{W}(s) = 4\pi\left(1 + \frac{s}{2}\right) \tan\left(\frac{\pi s}{2}\right)$$; all physically admissible perturbations decay exponentially or as damped oscillations, depending on the velocity parameter $V=x(\tau/t)$ defined through the steepest descent analysis of the inverse Mellin transform. For $V\geq V_0=2\pi$, decay is purely exponential; for smaller $V$, oscillations can accompany decay.
• 3D ($h=-1/2$): The analysis employs Balk–Zakharov (BZ) theory, involving winding number calculations of $\mathcal{W}(s)$. The key stability criterion is that the interval $[\sigma_-, \sigma_+]$ (maximal zero winding number interval in $s$) satisfies $\sigma_- + h < 0 \leq \sigma_+$. For 3D acoustics this holds, ensuring nonlinear and spectral stability (Costa et al., 13 Aug 2025).

In both cases, stability of both KZ and RJ states is established for small isotropic perturbations, but the mode of decay and spectral propagation differ by dimension and cascade direction.

Phenomenology of Perturbation Evolution

For KZ solutions, perturbations often propagate in $k$-space (logarithmic variable $x = \log(k/k_0)$) as damped, self-similar “fronts” or “peaks” with dimension-dependent propagation velocities and attenuation rates. In 2D, explicit damping and oscillatory patterns are observed; in 3D, BZ theory predicts both forward and backward cascading components, with distinct self-similar structures.

3. Dynamical Cascades and Generalized Solutions

Typical Kolmogorov–Zakharov spectra are stationary. However, studies of the 3-wave kinetic equations have demonstrated a class of time-dependent, energy-conserving weak solutions wherein energy originally distributed over finite $k$ cascades irreversibly to $k \to \infty$, ultimately concentrating as a Dirac mass at infinity (Soffer et al., 2018). In this scenario,

$$f(t,p)\,p\,d\mu(p) \rightharpoonup E\,\delta_{\{p=\infty\}}\quad\text{as }t\to\infty,$$

with $E$ total energy. This result provides a rigorous mathematical embodiment for the irreversible nature of the energy cascade in decaying turbulence for quadratic (3-wave) systems.

Such time-dependent solutions go beyond stationary KZ spectra, highlighting the importance of the kinetic equation's extended state space and providing a model for the gelation-like accumulation of energy at infinitely fine scales, a phenomenon not captured by stationary KZ theory.

4. Physical Realizability and the Role of Dissipation

Experimental realizations of KZ spectra require careful separation of injection and dissipation scales to create a conservative inertial range. In flexural wave turbulence (e.g., thin elastic plates), the Föppl–von Kármán equations together with weak turbulence theory predict the KZ spectrum

$E_k^{(2D)} = C\,\phi^{1/3}\ln^{1/3}(k_*/k),$

however, realistic damping $\gamma_k = a + b k^2$ present in experiments introduces significant dissipation within the inertial range, leading to nonconstant energy flux, steeper spectra, and deviations from the predicted scaling ($\sim \phi^{0.7}$ experimentally, versus $\phi^{1/3}$ by theory) (Miquel et al., 2014). Numerical simulations confirm that only when dissipation is artificially shifted outside the inertial range does the KZ spectrum emerge, with experimental and numerical spectra collapsing onto the theoretical master curve under such conditions. This demonstrates that the spectral distribution of dissipation fundamentally controls the ability to observe KZ regimes in practice.

5. KZ Framework in Applied and Statistical Contexts

The KZ kinetic framework plays a central role in applications beyond fundamental turbulence:

• Optical Fiber Communication: The KZ model is applied for energy redistribution among Fourier modes in dispersive nonlinear Schrödinger systems. A kinetic equation is derived by evolving the power spectral density (PSD) using cumulant expansions, classifying resonance quartets, and improving over the Gaussian Noise (GN) model by accurately preserving energy and higher-order statistical correlations. This yields more robust predictions of nonlinear interference and extends naturally to non-stationary inputs and multi-span fibers (Yousefi, 2014).
• Numerical Simulation of Wave Turbulence: Direct confirmation of KZ scaling in inertial wave turbulence (e.g., rotating fluids) is achieved by targeting wave-resonant modes in simulations, confirming the $$\mathcal{E}(k_\perp, k_\parallel) \propto k_\perp^{-1/2} k_\parallel^{-5/2}$$ scaling (Reun et al., 2020), matching analytic KZ predictions [cf. Galtier et al.].
• Nonlinear Stationary State Construction: Recent advances provide explicit constructions and stability analysis of out-of-equilibrium steady states with constant energy (direct) or mass (inverse) cascades in the kinetic wave equation, employing fixed point arguments, Mellin analysis, and careful handling of mass/energy fluxes and neutral modes (Collot et al., 2022).

6. Broader Theoretical Implications

Contemporary KZ theory is supplemented by rigorous uniqueness properties (e.g., nonexistence of compactly supported nontrivial solutions at two times for dispersive equations, as established via Carleman estimates (Bustamante et al., 2010)) and strong connections to concentration-compactness, profile decomposition, and refined dispersive estimates for minimal and non-scattering solutions in nonlinear dispersive PDEs (e.g., energy-critical Zakharov systems in 4D) (Candy, 2022).

The dynamical and statistical behaviors captured by KZ solutions frame ongoing research into transitions between regular and singular regimes (e.g., blow-up versus KZ spectra (Cher et al., 2019)), the influence of stochastic effects and noise regularization (Herr et al., 2023), and the extension of deterministic techniques (e.g., normal forms, local smoothing) to SPDEs.

7. Summary Table: Stability of KZ and RJ States in Acoustic Turbulence

Spectrum	Dimensionality	Perturbation Evolution	Stability Mechanism
KZ (Kolmogorov–Zak.)	2D	Exponential/oscillatory decay	Mellin transform/Carleman
KZ (Kolmogorov–Zak.)	3D	Self-similar propagation (forward/backward cascades)	Balk–Zakharov winding number criterion
RJ (Rayleigh–Jeans)	all	Pure exponential decay	Direct analysis (ultraviolet cutoff)

In summary, Kolmogorov–Zakharov solutions are the fundamental stationary (and sometimes dynamical) scaling laws for turbulent spectral transfer in weakly nonlinear wave systems. They are mathematically characterized by precise power-law exponents, stabilized or destabilized by dimension and collision kernel structure, and their physical realization is sensitive to the distribution of physical dissipation and the presence of inertial ranges. Recent analyses provide not only proofs of their stability and uniqueness but also new forms of dynamical and generalized (weak) solutions that go beyond stationary cascades, enriching the theoretical landscape of wave turbulence (Costa et al., 13 Aug 2025, Collot et al., 2022, Soffer et al., 2018, Miquel et al., 2014, Yousefi, 2014, Reun et al., 2020).