Bidirectional shallow-water wave turbulence

We study bidirectional one-dimensional (1-D) shallow-water waves within a class of Boussinesq equations, including the integrable Kaup–Boussinesq (KB) equation and a truncated-dispersion variant, which serves as a representative non-integrable model. For these two systems, the normal-form transformation yields an interaction coefficient of the same general structure, differing only through the dispersion relation. We derive this coefficient and numerically confirm that it vanishes on the resonant manifold for the KB equation, as expected from integrability. In contrast, the non-integrable model admits a non-vanishing interaction coefficient consistent with broken integrability, producing a non-trivial wave kinetic equation (WKE), which is the first known in a 1-D shallow-water setting. The resulting WKE is non-homogeneous in nature due to the non-homogeneity of the corresponding dispersion relation; however, approximate Kolmogorov–Zakharov solutions can be derived in a novel way under certain approximations. Numerical experiments in two settings validate the kinetic predictions and elucidate the underlying dynamics: (i) in free-evolution cases of the KB equation, despite complete integrability and the invariance of the discrete nonlinear spectrum guaranteed by isospectrality, an initial arbitrary wavenumber spectrum undergoes substantial evolution driven by quasi-resonant triad interactions and (ii) in forced-dissipated cases of the non-integrable equation, we find stationary power-law spectra that agree with the theoretical predictions.