Longons from the nonlinear dispersion of Galerkin regularization

Abstract

Irregular compactons and peakons from some nonlinear dispersions can be regularized by another type of nonlinear dispersion, defined by a pseudo-differential operator in physical space for the Galerkin truncation preserving finite Fourier modes of wavenumbers no larger than $K$. This resembles yet differs from the Korteweg-de Vries (KdV) regularization of the Burgers-Hopf (BH) equation. The Galerkin-regularized compacton, peakon, KdV, and BH dynamics exhibit novel traveling waves and interacting solitonic structures amidst weaker, less-ordered components (`longons'). Quasi-periodic solutions are also constructed with on-torus invariants, towards a potential Kolmogorov-Arnold-Moser (KAM) argument with presumably whiskered tori. The latter are persistent against the approximation of the truncation by a type of linear dispersion models, resulting in similar longulent states. Time-dependent and stationary behaviors in the large-$K$ limit are addressed with numerical results.