Turbulent cascades in a truncation of the cubic Szegő equation and related systems

Abstract

We introduce a truncated version of the cubic Szegő equation, an integrable model for deterministic turbulence. In this truncation, a majority of the Fourier mode couplings are eliminated, while the signature features of the model are preserved, namely, a Lax pair structure and a hierarchy of finite-dimensional dynamically invariant manifolds. Despite the impoverished structure of the interactions, the turbulent behaviors of our new equation are stronger in an appropriate sense than for the original cubic Szegő equation. We construct explicit analytic solutions displaying exponential growth of Sobolev norms. We furthermore introduce a family of models that interpolate between our truncated system and the original cubic Szegő equation, along with other related deformations. These models possess Lax pairs and invariant manifolds, and display a variety of turbulent cascades. We additionally mention numerical evidence, in some related systems, for an even stronger type of turbulence in the form of a finite-time blow-up.