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

TL;DR

This paper explores complex wave states in Galerkin-regularized hydrodynamic systems, identifying solitonic structures called longons and their stability, contributing to understanding pseudo-integrability in such models.

Contribution

It introduces the concept of longulent states with solitonic longons and develops on-torus invariants for constructing KAM tori, advancing pseudo-integrability theory.

Findings

Identification of longulent states with solitonic structures

Numerical evidence of persistence under perturbations

Development of on-torus invariants for KAM tori

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.