Travelling-wave, Quasi-periodic, and Longulent States of the Galerkin-regularized Hydrodynamic-type Systems

Abstract

Travelling-wave, quasi-periodic and ``longulent'' states of the Galerkin-regularized systems preserving finite Fourier modes are exposed. The longulent states are characterized by solitonic structures, called ``longons'', accompanied by disordered components, which is associated to whiskered tori according to the \textit{a-posteriori} Kolmogorov-Arnold-Moser (KAM) theorem. On-torus invariants are introduced for constructing the KAM tori, towards a potential pseudo-integrability theory. Persistence of the longulent states with respect to certain dispersive and dissipative perturbations are also numerically indicated.