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Stability of Periodic Waves of 1D Cubic Nonlinear Schrödinger Equations

Stability of Periodic Waves of 1D Cubic Nonlinear Schrödinger Equations AbstractWe study the stability of the cnoidal, dnoidal and snoidal elliptic functions as spatially-periodic standing wave solutions of the 1D cubic nonlinear Schrödinger 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 (in a certain parameter range) and snoidal waves against same-period perturbations, thus providing an alternate proof of this (known) fact, 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 Schrödinger 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics Research Express Oxford University Press

Stability of Periodic Waves of 1D Cubic Nonlinear Schrödinger Equations

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References (33)

Publisher
Oxford University Press
Copyright
© The Author(s) 2017. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com.
ISSN
1687-1200
eISSN
1687-1197
DOI
10.1093/amrx/abx004
Publisher site
See Article on Publisher Site

Abstract

AbstractWe study the stability of the cnoidal, dnoidal and snoidal elliptic functions as spatially-periodic standing wave solutions of the 1D cubic nonlinear Schrödinger 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 (in a certain parameter range) and snoidal waves against same-period perturbations, thus providing an alternate proof of this (known) fact, 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 Schrödinger 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.

Journal

Applied Mathematics Research ExpressOxford University Press

Published: Sep 1, 2017

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