Abstract
An extension to wave turbulence theory shows that in a system of nonlinear waves, correlations among the waves arise, causing ``ghost'' excitations that lead to coherent structures in physical space.
Highlights
Wave turbulence theory has led to successful predictions on the wave spectrum in many fields of physics [1,2]
We show that Hamiltonian nonlinear dispersive wave systems with cubic nonlinearity and random initial data develop, during their evolution, anomalous correlators
The standard object to look at is the second-order correlator, hakðtÞaÃl ðtÞi, where hÁ Á Ái is an average over an ensemble of initial conditions with different random phases and amplitudes
Summary
Wave turbulence theory has led to successful predictions on the wave spectrum in many fields of physics [1,2]. After multiplication by its complex conjugate and taking averages over different realizations with the same statistics, we get NðaÞðk; ΩÞ 1⁄4 nðaÞðk; t0ÞδðΩ − ωkÞ; ð13Þ where nðaÞðk; t0Þ is the standard wave spectrum at time t0 related to the second-order correlator as haðki; t0Þaðki; t0ÞÃi 1⁄4 nðaÞðki; t0Þδðki − kjÞ; ð14Þ and defined via the autocorrelation function as nðaÞðki; t0Þ. This is related to the fact that the amplitude for each wave number is not constant in time; the amplitude-dependent frequencies are not constant in time and they oscillate around a mean value with some fluctuations Those results are well understood, at least in the weakly nonlinear regime, and can be predicted using wave turbulence tools; see Refs. As can be seen from the plot, there is a monotonic growth of the ghost waves that, for very large nonlinearity, can reach values up to 25% of the total number
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