Numerical simulation of nonlinear dispersive quantization

When posed on a periodic domain in one space variable, linear dispersive evolution equations with integral polynomial dispersion relations exhibit strikingly different behaviors depending upon whether the time is rational or irrational relative to the length of the interval, thus producing the Talbot effect of dispersive quantization and fractalization. The goal here is to show that these remarkable phenomena extend to nonlinear dispersive evolution equations. We will present numerical simulations, based on operator splitting methods, of the nonlinear Schrödinger and Korteweg-deVries equations with step function initial data and periodic boundary conditions. For the integrable nonlinear Schrödinger equation, our observations have been rigorously confirmed in a recent paper of Erdoǧan and Tzirakis, [10].