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We show that Hamiltonian nonlinear dispersive wave systems with cubicnonlinearity and random initial data develop, during their evolution, anomalouscorrelators. These are responsible for the appearance of "ghost" excitations,i.e. those characterized by negative frequencies, in addition to the positiveones predicted by the linear dispersion relation. We use generalization of theWick's decomposition and the wave turbulence theory to explain theoreticallythe existence of anomalous correlators. We test our theory on the celebrated$\beta$-Fermi-Pasta-Ulam-Tsingou chain and show that numerically measuredvalues of the anomalous correlators agree, in the weakly nonlinear regime, withour analytical predictions. We also predict that similar phenomena will occurin other nonlinear systems dominated by nonlinear interactions, includingsurface gravity waves. Our results pave the road to study phase correlations inthe Fourier space for weakly nonlinear dispersive wave systems.

Anomalous correlators, "ghost" waves and nonlinear standing waves in the $\beta$-FPUT system