Dispersive quantization and fractalization for multi-component dispersive equations

Abstract

In this paper, the dispersive quantization and fractalization phenomena for the multi-component systems of the dispersive evolution equations on a bounded interval subject to periodic boundary conditions and discontinuous initial profiles are investigated. Firstly, for a class of two-component linear system of dispersive evolution equations, the dispersive quantization conditions are provided. It is shown that for those systems which satisfy the conditions, the evolution of the step or more general discontinuous functions lead to the quantized structures at rational times. Next, numerical experiments, based on the Fourier spectral method, suggest that such effects persist into the nonlinear regime, as long as the associated linearization satisfies the dispersive quantization conditions. It turns out that the extra component in the multi-component system may break down the structure of dispersive quantization and consequently evanish the quantization phenomena even at the rational times.