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Alternating the signs of the frequencies for Fourier components with even andodd (normalized) wavenumbers in the Korteweg-de Vries (KdV) equation maintainsa dispersive shock with two-sided similar oscillations (like some plasma andquantum shocks). Such a modification, referred to as the even-odd-alternatingKdV (aKdV) model, can also lead to fractalization and quantum revival (Talboteffect). Further refinement to introduce "boy-girl-twin" frequencies eliminatesthe disturbances on shock propagation caused by the even-odd asymmetry in thedispersion. Solitonic structures, including the shock as a "shocliton," areprominently present. The traveling-wave solutions and on-torus invariants forquasi-periodic tori, for the Hamiltonian dynamics with the integrabilitystructures broken (such as the loss of infinitely many invariants), areproposed to account for the (pseudo-)periodicity in the aKdV space-timepatterns which correspond to presumably whiskered tori.

Even-odd alternative dispersions and beyond: Part I. Close oscillations on both sides of the (anti-)shock, quantum revival and fractalization