Construct GrCGL longulent states from GrNLS-based initial data

Develop a method to construct nontrivial longulent states in the Galerkin-regularized complex Ginzburg-Landau (GrCGL) equation starting from corresponding initial data derived from Galerkin-regularized nonlinear Schrödinger (GrNLS) solutions, such as multi-frequency quasi-periodic or longulent configurations.

Background

Motivated by GrNLS results, the authors attempt to transfer quasi-periodic and longulent structures to GrCGL by suitable initial data. Numerical experiments reported in the paper often converge to clean periodic travelling waves, not exhibiting nontrivial longulence under GrCGL dynamics.

They explicitly state that they have not yet been able to construct GrCGL longulent states from such GrNLS-derived data, suggesting the need for new techniques (e.g., Lyapunov methods, tailored torus-specific invariants) to achieve persistence or emergence of longulence under complex Ginzburg-Landau dynamics.