Coherent energy cascades in nonlinear disordered Hamiltonians

Abstract

The familiar wave kinetic theory revolves around turbulent energy cascades characterized by random phases of the individual normal modes. By contrast, recent years have revealed turbulent cascades where the transfer of energy is mediated by phase-coherent dynamics. Such coherent cascades have been previously established in deterministic systems with a tightly controlled algebraic structure of the mode interactions, such as nonlinear Schrödinger equations. Here, we show that the presence of phase-coherent cascades is robust with respect to substantial alterations of the mode couplings. We prove that, in infinite-dimensional resonant Hamiltonian systems with fully resonant normal mode spectra and random nonlinear couplings, a structured subset of couplings --- an infinitesimal fraction of the total --- suffices to support coherent cascades characterized by phase alignment, the formation of power-law spectra, and either unbounded growth of Sobolev norms or a finite-time blow-up. Numerical simulations further demonstrate the emergence of coherent cascades from random initial data through a process of phase locking, suggesting a mechanism for their emergence in broader classes of initial data and systems. We thus show that disordered mode interactions can sustain, rather than block, coherent energy transport across arbitrarily distant scales.