Inverse cascade anomalies in fourth-order Leith models

We analyze a family of fourth-order non-linear diffusion models corresponding to local approximations of four-wave kinetic equations of weak wave turbulence. We focus on a class of parameters for which a dual cascade behavior 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.