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Stability of periodic waves for the fractional KdV and NLS equations

Part of: Spectral theory and eigenvalue problems Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  12 August 2020

Sevdzhan Hakkaev  and
Atanas G. Stefanov
Sevdzhan Hakkaev
Affiliation:
Department of Mathematics and Computer Science, Istanbul Aydin University, Istanbul, Turkey Faculty of Mathematics and Informatics, Shumen University, Shumen, Bulgaria (sevdzhanhakkaev@aydin.edu.tr)
Atanas G. Stefanov
Affiliation:
Department of Mathematics, University of Kansas, 1460 Jayhawk Boulevard, Lawrence KS 66045–7523, USA (stefanov@ku.edu)
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Abstract

We consider the focussing fractional periodic Korteweg–deVries (fKdV) and fractional periodic non-linear Schrödinger equations (fNLS) equations, with L2 sub-critical dispersion. In particular, this covers the case of the periodic KdV and Benjamin-Ono models. We construct two parameter family of bell-shaped travelling waves for KdV (standing waves for NLS), which are constrained minimizers of the Hamiltonian. We show in particular that for each $\lambda > 0$, there is a travelling wave solution to fKdV and fNLS $\phi : \|\phi \|_{L^2[-T,T]}^2=\lambda $, which is non-degenerate. We also show that the waves are spectrally stable and orbitally stable, provided the Cauchy problem is locally well-posed in Hα/2[ − T, T] and a natural technical condition. This is done rigorously, without any a priori assumptions on the smoothness of the waves or the Lagrange multipliers.

Keywords

Fractional KdVfactional NLSperiodic wavesstability

MSC classification

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger)
Secondary: 35P10: Completeness of eigenfunctions, eigenfunction expansions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 4 , August 2021 , pp. 1171 - 1203
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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