We show that Hamiltonian nonlinear dispersive wave systems with cubic nonlinearity and random initial data develop, during their evolution, anomalous correlators. These are responsible for the appearance of "ghost" excitations, i.e. those characterized by negative frequencies, in addition to the positive ones predicted by the linear dispersion relation. We use generalization of the Wick’s decomposition and the wave turbulence theory to explain theoretically the existence of anomalous correlators. We test our theory on the celebrated $\beta$-Fermi-Pasta-Ulam-Tsingou chain and show that numerically measured values of the anomalous correlators agree, in the weakly nonlinear regime, with our analytical predictions. We also predict that similar phenomena will occur in other nonlinear systems dominated by nonlinear interactions, including surface gravity waves. Our results pave the road to study phase correlations in the Fourier space for weakly nonlinear dispersive wave systems.

中文翻译：

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非线性色散波系统中的异常相关器

我们表明，具有三次非线性和随机初始数据的哈密顿非线性色散波系统在其演化过程中发展了异常相关器。除了线性色散关系预测的正向激发外，这些还负责出现“重影”激发，即以负频率为特征的激发。我们使用维克分解的泛化和波湍流理论从理论上解释异常相关器的存在。我们对著名的理论进行检验$\beta$-Fermi-Pasta-Ulam-Tsingou链，并表明在弱非线性条件下，异常相关器的数值测量值与我们的分析预测一致。我们还预测相似的现象将在其他以非线性相互作用为主的非线性系统中发生，包括表面重力波。我们的结果为研究弱非线性色散波系统的傅立叶空间中的相位相关性铺平了道路。