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 when the time is rational or irrational relative to the length of the interval: the Talbot phenomenon of dispersive quantization and fractalization. The purpose of this paper is to show that these phenomena extend to nonlinear dispersive evolution equations. We will present numerical simulations of the integrable nonlinear Schrödinger equation and the nonlinear Korteweg–deVries equation with step function initial data and periodic boundary conditions. Convergence of our numerical scheme, which is based on operator splitting methods, will be proved for the nonlinear Schrödinger equation, while the difficulties in the proof of convergence for the Korteweg–deVries equation will be explained.