MIT PDE/Analysis Seminar

Abstract: The problem of Sobolev norm growth—namely, the transfer of energy from low to arbitrarily high modes—has been extensively studied in Hamiltonian systems with a deterministic structure, such as the cubic nonlinear Schrödinger equation. In this talk, I will present an extension of this problem to Hamiltonian systems dominated by random nonlinear interactions. I will introduce analytic solutions describing three types of energy cascades that lead either to unbounded growth or to finite-time blow-up of Sobolev norms. After, I will present numerical simulations demonstrating the rapid emergence of these dynamics from incoherent initial conditions. Taken together, these results demonstrate coherent energy cascades as robust mechanisms of energy transfer in some systems with random structures.