双向浅水波湍流
Bidirectional shallow-water wave turbulence
AI总结：
本文研究了双向一维浅水波在Boussinesq方程类中的行为，通过正常形式变换得出交互系数，发现可积的Kaup-Boussinesq方程中交互系数为零，而非可积模型产生非平凡波动能方程。
摘要：
我们研究了Boussinesq方程类中的一维双向浅水波，包括可积的Kaup-Boussinesq（KB）方程和一个截断色散变体，后者作为非可积模型的代表。对于这两个系统，正常形式变换得到的交互系数具有相同的通用结构，仅通过色散关系不同而区分。我们推导出该系数，并数值验证其在KB方程的共振流形上为零，如文献所预期。相比之下，非可积模型具有非零的交互系数，产生非平凡的波动能方程（WKE），这是在一维浅水设置中首次已知的。由此产生的WKE本质上是非均质的，由于对应的色散关系非均质；然而，在某些近似下，可以以新方法导出近似Kolmogorov-Zakharov（KZ）解。在两种设置中的数值实验验证了动能预测并阐明了底层动力学：（i）在KB方程的自由演化情况下，尽管完全可积且由等谱性保证的离散非线性谱不变，任意初始波数谱却受到准共振三元组相互作用的显著驱动；（ii）在非可积方程的强迫耗散情况下，我们发现与理论预测一致的稳态幂律谱。
英文摘要：
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 in the literature. In contrast, the non-integrable model admits a non-vanishing interaction coefficient, 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 Kolomogrov-Zakharov (KZ) 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; (ii) in forced-dissipated cases of the non-integrable equation, we find stationary power-law spectra that agree with the theoretical predictions.
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