Symmetry underlying organized pulse–wiggle patterns

Determine the symmetry, if any, that organizes the strong solitonic pulses ("longons") and weaker wiggles observed in the evolving wave solutions of Galerkin-regularized hydrodynamic-type systems on a 2π-periodic domain, particularly during the developing phases before maturation into solitonic longons.

Background

The paper studies travelling-wave and quasi-periodic solutions, as well as robust longulent states, in several Galerkin-regularized hydrodynamic-type models (including Burgers–Hopf, KdV, and compacton/peakon variants). Numerical experiments reveal strong, solitonic pulses accompanied by weaker, less ordered components. In the developing phases of these solutions, the authors observe patterns suggesting an underlying organizing principle.

They explicitly note that the symmetry responsible for these organized structures is unknown, highlighting an unresolved structural property of the solutions across the examined Galerkin-regularized systems.