Constructing Longulence in the Galerkin-Regularized Nonlinear Schrödinger and Complex Ginzburg-Landau Systems

(Quasi-)periodic solutions are constructed analytically for Galerkin-regularized or truncated nonlinear Schrödinger (GrNLS) systems preserving finite Fourier freedoms. GrNLS admits travelling-wave or multi-phase solutions, including monochromatic solutions independent of the truncation and quasi-periodic ones with or without additional on-torus invariants. Numerical tests show that instability leads such solutions to nontrivial longulent states with remarkable solitonic structures (called “longons”) admist disordered weaker components, corresponding to presumably whiskered tori. In the strong-coupling limit (e.g., the self-phase modulation equation in optics), neutral stability holds for the condensates, without the modulational instability, but not generally for other multi-phase (quasi-)periodic solutions from some of which the longulent state developed is also adressed. The possibility of nontrivial Galerkin-regularized complex Ginzburg-Landau longulent states is also discussed for motivation.