Stability of Periodic Waves of 1D Cubic Nonlinear Schrödinger Equations

Abstract

We study the stability of the cnoidal, dnoidal and snoidal elliptic functions as spatially-periodic standing wave solutions of the 1D cubic nonlinear Schrodinger equations. First, we give global variational characterizations of each of these periodic waves, which in particular provide alternate proofs of their orbital stability with respect to same-period perturbations, restricted to certain subspaces. Second, we prove the spectral stability of the cnoidal waves against same-period perturbations (in a certain parameter range), and provide an alternate proof of this (known) fact for the snoidal waves, which does not rely on complete integrability. Third, we give a rigorous version of a formal asymptotic calculation of Rowlands to establish the instability of a class of real-valued periodic waves in 1D, which includes the cnoidal waves of the 1D cubic focusing nonlinear Schrodinger equation, against perturbations with period a large multiple of their fundamental period. Finally, we develop a numerical method to compute the minimizers of the energy with fixed mass and momentum constraints. Numerical experiments support and complete our analytical results.