Modulation Equations for Nearly Integrable PDEs with Periodic Boundary Conditions

Abstract

This note, following my lecture, is a summary of the current state of practical application of periodic, N-phase modulation equations. First, I sketched interesting bifurcation phenomena that have been documented in [1] through extensive direct numerical investigations. These long-lived “attractors” of the damped, driven periodic sine-Gordon equation consist of various flows among coherent spatial structures. The temporal behavior ranges from periodic to quasiperiodic to chaotic, quantified in terms of several diagnostics. The spatial resolution during each flow is coherent and low-dimensional. In addition to Fourier projections, we use the sine-Gordon spectral transform to independently measure the “N-phase integrable mode approximations” of the perturbed field at each time step. The upshot is that the attractors are well-approximated, at least locally, by modulated N-phase periodic waves, where N remains small into the chaotic regime.