ABOUT PLANE PERIODIC WAVES OF THE NONLINEAR SCHRODINGER EQUATIONS

The present contribution contains a quite extensive theory for the stability analysis of plane periodic waves of general Schrodinger equations. On the one hand, we put the one-dimensional theory, or in other words the stability theory for longitudinal perturbations, on par with the one available for systems of Korteweg type, including results on coperiodic spectral instability, nonlinear coperiodic orbital stability, sideband spectral instability and linearized large-time dynamics in relation with modulation theory, and resolutions of all the involved assumptions in both the small-amplitude and large-period regimes. On the other hand, we provide extensions of the spectral part of the latter to the multidimensional context. Notably, we provide suitable multidimensional modulation formal asymptotics, validate those at the spectral level, and use them to prove that waves are always spectrally unstable in both the small-amplitude and the large-period regimes.