Dispersion of discontinuous periodic waves

Abstract

The dynamic evolution of linearly dispersive waves on periodic domains with discontinuous initial profiles is shown to depend remarkedly upon the asymptotics of the dispersion relation at large wavenumbers. Asymptotically linear or sublinear dispersion relations produce slowly changing waves, while those with polynomial growth exhibit dispersive quantization, a.k.a. the Talbot effect, being (approximately) quantized at rational times, but a non-differentiable fractal at irrational times. Numerical experiments suggest that such effects persist into the nonlinear regime, for both integrable and non-integrable systems. Implications for the successful modelling of wave phenomena on bounded domains and numerical challenges are discussed.