Even-odd paired dispersion for shocks with two-sided oscillations: the (modified) Korteweg-de Vries equation

Jian-Zhou Zhu

Published 2023-02-22Version 1

Our curiosity of the structural nature of the dynamics was motivated by two observations: mathematically, the variational principle and Hamiltonian formulation of some models, such as the Korteweg-de Vries (KdV) equation, are preserved, \textit{mutatis mutandis}, if each mode of freedom is assigned a different dispersion coefficient; and, physically, some oscillations on both sides of some ion-acoustic and quantum shocks. We thus propose re-grouping and re-assignment, particularly the distinguishment and pairing of opposite signs for alternative Fourier components, of the dispersions for different dynamical modes. The KdV equation with periodic boundary condition and longest-wave sinusoidal initial field, as used by N. Zabusky and M. D. Kruskal, is chosen for our case study with such pairing of the Fourier modes of (normalized) even and odd wavenumbers. Numerical results verify the two-sided oscillations around the shock but also indicate even more, including non-convergence to the classical shock (described by the entropy solution), non-thermalization and applicability to other models including the modified KdV equation with cubic nonlinearity. A unification of various dispersive models, keeping the essential mathematical elegance of each, for phenomena with complicated dispersion relation is thus suggested.