Stability of periodic waves of 1D cubic nonlinear Schr{\\"o}dinger\n equations
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Bibliographic record
Abstract
We study the stability of the cnoidal, dnoidal and snoidal elliptic functions\nas spatially-periodic standing wave solutions of the 1D cubic nonlinear\nSchr{\\"o}dinger equations. First, we give global variational characterizations\nof each of these periodic waves, which in particular provide alternate proofs\nof their orbital stability with respect to same-period perturbations,\nrestricted to certain subspaces. Second, we prove the spectral stability of the\ncnoidal waves against same-period perturbations (in a certain parameter range),\nand provide an alternate proof of this (known) fact for the snoidal waves,\nwhich does not rely on complete integrability. Third, we give a rigorous\nversion of a formal asymptotic calculation of Rowlands to establish the\ninstability of a class of real-valued periodic waves in 1D, which includes the\ncnoidal waves of the 1D cubic focusing nonlinear Schr{\\"o}dinger equation,\nagainst perturbations with period a large multiple of their fundamental period.\nFinally, we develop a numerical method to compute the minimizers of the energy\nwith fixed mass and momentum constraints. Numerical experiments support and\ncomplete our analytical results.\n
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