Wave turbulence on rational and irrational tori

We study wave turbulence in the nonlinear Schrödinger equation on two-dimensional domains of rational and irrational aspect ratios (i.e. rational and irrational tori). Situations with and without high-frequency dissipation are considered. For the former, we numerically study the quasi-stationary power-law spectra and energy cascades. We show that these quantities are remarkably different on the two tori, with reasons explained in terms of the discrete resonant manifold. For the latter, the focus is on the migration of initial data to high frequencies measured by the Sobolev norm. We will review the study of the problem in the harmonic analysis community, and discuss our new mathematical proof on the existence of a barrier to the energy cascade on irrational tori, followed by numerical demonstrations. The results in this work have important implications in understanding simulations of wave turbulence in periodic domains.