Wave kinetics in an integrable model -- the Kaup-Boussinesq system

TL;DR

This work challenges the conventional view that integrable systems cannot support wave turbulence by deriving a nontrivial wave kinetic equation (WKE) for the completely integrable Kaup-Boussinesq (KB) system. Using a normal-form transformation to remove quadratic terms, the authors obtain a quartic interaction with a nonzero four-wave kernel on the resonant manifold, and they show the WKE is inherently nonhomogeneous due to KB's dispersion. They compute the interaction kernel $T_{k123}$ and, because of the inhomogeneous dispersion, derive two distinct Kolmogorov-Zakharov spectra: $n_k\sim \kappa^{-3}$ (Case A) and $n_k\sim \kappa^{-3/2}$ (Case B), each with its own locality windows. Numerical simulations of free evolution reveal rapid thermalization of the wave spectrum while the exact soliton spectrum remains isospectral, and forced-dissipative runs reproduce the predicted power-law ranges, confirming nontrivial wave kinetics in an integrable framework and highlighting a nuanced link between integrability and kinetic theory.

Abstract

We study wave turbulence in one-dimensional (1-D) bidirectional shallow water waves described by the Kaup-Boussinesq (KB) equation, which is known to be an integrable system. In contrast to the generally accepted empirical belief that an integrable system yields no kinetic theory, we derive and validate a non-trivial wave kinetic equation (WKE) for the KB system with a non-zero interaction coefficient on the four-wave resonant manifold. This WKE is non-homogeneous in nature due to the non-homogeneity in the dispersion relation of the KB system; however, approximate Kolomogrov-Zakharov (KZ) solutions can be derived in a novel way under certain approximations. We numerically verify the theoretical findings in two cases: (i) In free-evolution cases, although the discrete (nonlinear) spectrum remains unchanged as guaranteed by an integrable system's isospectrality, an initial arbitrary wavenumber spectrum quickly evolves into a thermo-equilibrium state, demonstrating the kinetic aspect of the system; (ii) in forced-dissipated cases, we find stationary power-law spectra that agree with the theoretical predictions.

Figures (15)

• Figure 1: Initial (solid gray) and stationary (solid blue) spectra at $t=0$ and $t=2\times10^6T_p$ for the free-evolution case with $\alpha=\beta=0.2$. The thermal-equilibrium scaling is denoted (dashed black line).
• Figure 2: The spectrum of discrete eigenvalues $\zeta$ (blue and gray points) obtained via DST at (a) $t=0$ and (b) $t\approx2\times10^6 T_p$, corresponding to the times of initial and stationary spectra in Fig.\ref{['fig:nk_zeta_spec']}. The lower half $E$-plane (i.e., where $\textnormal{Im}[E(\zeta)]<0$) is marked by the red hashed region and the continuous spectrum region (i.e., where $\textnormal{Im}[E(\zeta)]=0$) is marked by the dashed black line.
• Figure 3: The initial spectrum (solid gray) and stationary spectra with $\alpha=0.2$ (dashed red), $\alpha=0.04$ (dashidot blue) and $\alpha=0.02$ (solid green), respectively taken at $5\times10^5T_p$, $8\times10^5T_p$, and $1.2\times10^6T_p$ . The thermal-equilibrium scaling is denoted (dashed black).
• Figure 4: The $k$-$\omega$ spectra $S_{\eta}(k,\omega)$ for (a) $\alpha=0.2$ and (b) $\alpha=0.02$. The KB dispersion relation \ref{['eq:disp']} is denoted by the dashed red line and the soliton dispersion relation (for the most dominant bound state) is denoted by the dashed black line in (a).
• Figure 5: Stationary wave action spectrum for Case A. The forcing range is marked by the dotted red lines and the theoretical KZ scaling $n_\kappa\sim \kappa^{-3}$ is denoted by the dashed black line.