Inverse cascade anomalies in fourth-order Leith models

Abstract

We analyze a family of fourth-order non-linear diffusion models corresponding to local approximations of 4-wave kinetic equations of weak wave turbulence. We focus on a class of parameters for which a dual cascade behaviour is expected with an infrared finite-time singularity associated to inverse transfer of waveaction. This case is relevant for wave turbulence arising in the Nonlinear Schrödinger model and for the gravitational waves in the Einstein’s vacuum field model. We show that inverse transfer is not described by a scaling of the constant-flux solution but has an anomalous scaling. We compute the anomalous exponents and analyze their origin using the theory of dynamical systems.

An effective semilocal model for wave turbulence in two-dimensional nonlinear optics

“…As for the boundary conditions, if one were concerned with a solution on the whole real line then physicality requires that the spectrum and fluxes vanish as ωfalse→normal∞. The situation as ωfalse→0 is quite involved, as one expects solutions that exhibit finite-time blowup there, associated with condensation [31]. At this point not only does the support of the spectrum violate the ω≫Λ condition, but the system becomes strongly nonlinear, meaning that the WKE (and hence the SLAM) no longer describe the dynamics.…”