Energy cascades and condensation via coherent dynamics in Hamiltonian systems

Abstract: This work makes analytic progress in the deterministic study of turbulence in Hamiltonian systems by identifying two types of energy cascade solutions and the corresponding large- and small-scale structures they generate. The first cascade represents condensate formation via a highly coherent process recently uncovered, while the second cascade, which has not been previously observed, leads to the formation of other large-scale structures. The concentration of energy at small scales is characterized in both cases by the development of a power-law spectrum in finite time, causing the blow-up of Sobolev norms and the formation of coherent structures at small scales. These structures approach two different types of singularities: a point discontinuity in one case and a cusp in the other. The results are fully analytic and explicit, based on two solvable families of Hamiltonian systems identified in this study.