Identification of the relevant stationary zero-λ solution for large truncation K

Identify, among the multiple stationary travelling-wave (λ=0) solutions of the Galerkin-regularized Burgers–Hopf system for large truncation K, the specific solution whose slight KAM-type deformation captures the observed longulent evolution, and assess the possibility that alternative stationary solutions arise when the right-hand side is replaced by i·ω_K(k)·û_k in the Fourier-mode equation.

Background

To connect longulent states with KAM tori, the authors consider stationary (λ=0) travelling-wave solutions u_sK of the Galerkin-regularized Burgers–Hopf system. They argue that, for certain initial data, a nearby stationary solution should underlie the longulent dynamics via a KAM deformation.

However, for large K there can be many stationary solutions, and the authors explicitly state that identifying the correct one remains unresolved. They also note that alternative stationary solutions may exist under a variant where the Fourier-mode right-hand side is replaced by i·ω_K(k)·û_k, further broadening the identification problem.