Staggered dispersions: Part I. Shocliton, quantum revival and fractalization

Alternating the signs of the frequencies for Fourier components with even and odd (normalized) wavenumbers in the nonlinear-wave, such as the Korteweg-de Vries, equations maintains a constantly drifting dispersive shock with similar solitonic oscillations on both sides, like some plasma and quantum shocks. Such $\mathit{\text{staggered}}\mathit{\text{dispersions}}$ support bi-directional waves and can symmetrize and deregularize (to some degree) the nonlinear dynamics, with physical consequences that can also be reflected in the fractalization and quantum revival (Talbot effect).