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

## TL;DR

This paper investigates the emergence of anomalous correlators and 'ghost' waves in the $eta$-FPUT system, revealing their role in nonlinear wave dynamics and extending understanding of phase correlations in dispersive systems.

## Contribution

It introduces a theoretical framework explaining anomalous correlators and 'ghost' waves, validated by numerical simulations in the $eta$-FPUT system, with implications for other nonlinear wave systems.

## Key findings

- Anomalous correlators develop during nonlinear evolution.
- Numerical results agree with analytical predictions in weakly nonlinear regime.
- Similar phenomena are predicted in other nonlinear dispersive systems.

## Abstract

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.