Existence of nontrivial longulent states in Galerkin-regularized CGL

Establish the existence of nontrivial longulent states in the Galerkin-regularized (Fourier-truncated) complex Ginzburg-Landau equation, characterized by persistent solitonic longons accompanied by a disordered component and significant truncation effects (i.e., not converging to untruncated CGL dynamics), potentially corresponding to whiskered invariant tori.

Background

The paper introduces "longulence" as statistically stable pseudo-periodic states featuring prominent solitonic structures (longons) within a disordered background, observed in Galerkin-regularized nonlinear Schrödinger (GrNLS) systems and associated with whiskered tori. While similar phenomena are sought in the Galerkin-regularized complex Ginzburg-Landau (GrCGL) setting, the authors emphasize that such states have not been proven to exist.

This problem concerns proving the existence of these nontrivial longulent states in GrCGL, which differ from trivial attractors (e.g., clean periodic travelling waves) and require genuine truncation effects and a disordered component. It aims to extend longulence beyond GrNLS to GrCGL and connect the dynamics to invariant tori in dissipative-complexified systems.