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Stability of periodic waves for the fractional KdV and NLS equations

Published online by Cambridge University Press:  12 August 2020

Sevdzhan Hakkaev
Department of Mathematics and Computer Science, Istanbul Aydin University, Istanbul, Turkey Faculty of Mathematics and Informatics, Shumen University, Shumen, Bulgaria (
Atanas G. Stefanov
Department of Mathematics, University of Kansas, 1460 Jayhawk Boulevard, Lawrence KS 66045–7523, USA (


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.

Research Article
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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