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The cubic Szego equation has been studied as an integrable model fordeterministic turbulence, starting with the foundational work of Gerard andGrellier. We introduce a truncated version of this equation, wherein a majorityof the Fourier mode couplings are eliminated while the signature features ofthe model are preserved, namely, a Lax-pair structure and a nested hierarchy offinite-dimensional dynamically invariant manifolds. Despite the impoverishedstructure of the interactions, the turbulent behaviors of our new equation arestronger in an appropriate sense than for the original cubic Szego equation. Weconstruct explicit analytic solutions displaying exponential growth of Sobolevnorms. We furthermore introduce a family of models that interpolate between ourtruncated system and the original cubic Szego equation, along with a few otherrelated deformations. All of these models possess Lax pairs, invariantmanifolds, and display a variety of turbulent cascades. We additionally mentionnumerical evidence that shows an even stronger type of turbulence in the formof a finite-time blow-up in some different, closely related dynamical systems.

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