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We have two basic observations: mathematically, the variational principle andHamiltonian formulation of some models, such as the Korteweg-de Vries (KdV)equation, are preserved, \textit{mutatis mutandis}, if each mode of freedom isassigned a different dispersion coefficient; and, physically, (similar)dispersive oscillations appear on both sides of some ion-acoustic and quantumshocks, not generated by the dynamics of the models such as the KdV(-Burgers)equation. We thus consider assigning different types of dispersions fordifferent dynamical modes, particularly with the alternation of the signs foralternative Fourier components, different to the two-sided KdV equations forhead-on collisions of nonlinear waves. The KdV equation with periodic boundarycondition and longest-wave sinusoidal initial field, as used by N. Zabusky andM. D. Kruskal, is chosen for our case study with such alternating-dispersion ofthe Fourier modes of (normalized) even and odd wavenumbers. Numerical resultsverify the capability of our model to produce two-sided (around the shock)similar oscillations and indicate even more, including the notion of(anti)shock-soliton duality and/or ``(anti)shocliton'', singularzero-dispersion limit or non-convergence to the classical shock (described bythe entropy solution) and non-thermalization (of the Galerkin-truncatedmodels). Extensions to other models and generalization of the mode-dependentdispersion models are also discussed, showcased respectively withBenjamin-Ono-type and the modified-KdV ones. A tentative physical applicationoriented towards modeling the ion-acoustic shock with the even-odd alternativedispersion model is made to compare against the traditional KdV-Burgersapproach.

Even-odd alternative dispersions and beyond. Part I. Oscillations on both sides of the (anti-)shock, shocliton and other indications