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 -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.
3 More- Received 21 January 2020
- Revised 21 March 2020
- Accepted 31 March 2020
DOI:https://doi.org/10.1103/PhysRevX.10.021043
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
When a large number of interacting waves are present in some system—such as the ocean or solar wind—it is no longer possible to describe waves individually. In such cases, physicists turn to “wave turbulence theory,” which is a mathematical tool for describing large ensembles of waves in a statistical sense. One of the assumptions of the theory is that waves that are initialized with uncorrelated phases remain uncorrelated as time evolves. While this is true in many circumstances, its validity is questioned for nonlinear systems, where the waves are described by nonlinear equations. By relaxing the assumption that waves remain uncorrelated, we have mathematically discovered intriguing new phenomena that can arise in nonlinear wave systems.
Our analysis shows that waves with positive wave numbers may experience strong correlations to corresponding waves with negative wave numbers, propagating in the opposite direction. Consequently, this leads to “ghost” excitations, wherein waves propagate in the opposite direction that they would propagate in if the nonlinearity in the system were removed. This corresponds to the formation of a standing-wave pattern in individual realizations of the system; these coherent structures exist despite the fact that, statistically, energy moves randomly between all modes of oscillation.
Our results are backed by simulations of the Fermi-Pasta-Ulam-Tsingou system, modeling a nonlinear string, but our analysis is applicable to a broad range of systems, including ocean waves and the nonlinear Schrödinger equation. Waves are universal in nature on all scales—from quantum mechanics to seismic waves—making nonlinear waves a vast and rich field and making our generalization of wave turbulence theory useful for a variety of physical applications.