Stochastic linearization of turbulent dynamics of dispersive waves in equilibrium and non-equilibrium state

Abstract

Characterizing dispersive wave turbulence in the long time dynamics is central to understanding of many natural phenomena, e.g., in atmosphere ocean dynamics, nonlinear optics, and plasma physics. Using the β-Fermi-Pasta-Ulam nonlinear system as a prototypical example, we show that in thermal equilibrium and non-equilibrium steady state the turbulent state even in the strongly nonlinear regime possesses an effective linear stochastic structure in renormalized normal variables. In this framework, we can well characterize the spatiotemporal dynamics, which are dominated by longwavelength renormalized waves. We further demonstrate that the energy flux is nearly saturated by the long-wavelength renormalized waves in non-equilibrium steady state. The scenario of such effective linear stochastic dynamics can be extended to study turbulent states in other nonlinear wave systems.

Effective dispersion in the focusing nonlinear Schrödinger equation

“…In particular, the actions should reduce to the mode amplitudes |a k n (0)| in the limit of small nonlinearity, and the frequencies of the angles to the frequencies in Eq. (8). In this respect, intuitively, both the results of Section III A 1 as well as those of Ref.…”

“…III B 4 and Fig. 12, we took q 0 = 1/ √ 2, ρ 0 + iφ 0 = −0.6909 − 1.4166 i, L = 2π, J = 2 8 and N x = 2 12 .…”

“…A description of this phenomenon at moderate to strong nonlinearities was provided in [3,4]. In these works, a renormalized, effective dispersion relation (EDR) for the β -Fermi-Pasta-Ulam (FPU) chain and its corresponding renormalized Hamiltonian were established (see also [7,8]). In [5], these concepts were extended to the one-dimensional defocusing Majda-McLaughlin-Tabak (MMT) model of wave turbulence [9][10][11].…”