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arXiv · 1606.04215

Stability of periodic waves of 1D cubic nonlinear Schr{ö}dinger equations

Stephen Gustafson·Stefan Le Coz·Tai-Peng Tsai
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Abstract

We study the stability of the cnoidal, dnoidal and snoidal elliptic functions as spatially-periodic standing wave solutions of the 1D cubic nonlinear Schr{ö}dinger 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 Schr{ö}dinger 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.

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Stephen Gustafson, Stefan Le Coz, Tai-Peng Tsai. 2016-10-12. Stability of periodic waves of 1D cubic nonlinear Schr{ö}dinger equations. https://arxiv.org/abs/1606.04215

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