Renormalized Dispersive Relations in Wave Turbulence

The β-Fermi-Pasta-Ulam (FPU) system has been under active investigation due to its rich dynamics and various relations to mathematical and physical theories. Here, we focus on the issue of whether there exists effective dispersive characteristics for this system. Using multiple scale analysis, we propose a theoretical framework to obtain various renormalized dispersion relations in thermal equilibrium states and explain their corresponding physical meanings. Surprisingly, these relations can be generalized to the non-equilibrium steady state with driving-damping in both real and Fourier space.

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 long-wavelength renormalized waves. The scenario of such effective linear stochastic dynamics can be extended to study turbulent states in other nonlinear wave systems.