Constructing longulence in the Galerkin-regularized nonlinear Schr\"odinger and complex Ginzburg-Landau systems

Abstract

(Quasi-)periodic solutions are constructed analytically for Galerkin-regularized or truncated nonlinear Schr\"odinger (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.