Travelling-Wave, Quasi-Periodic, and Persistent Longulent States of the Galerkin-Regularized Hydrodynamic-Type Systems

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 with the a-posteriori Kolmogorov-Arnold-Moser (KAM) theorem. On-torus invariants are introduced for constructing the KAM tori, towards a potential theory of pseudo-integrability in the sense of specifying precisely the corresponding whiskered tori. Persistence of the longulent states with respect to certain dispersion and dissipation (fine matched by a particularly designed driving model) perturbations are also suggested with numerical results.