Stability of periodic waves for the fractional KdV and NLS equations

HAKKAEV S. A., Stefanov A. G.

Proceedings of the Royal Society of Edinburgh Section A: Mathematics, cilt.151, sa.4, ss.1171-1203, 2021 (SCI-Expanded, Scopus) identifier identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 151 Sayı: 4
  • Basım Tarihi: 2021
  • Doi Numarası: 10.1017/prm.2020.54
  • Dergi Adı: Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Aerospace Database, Communication Abstracts, INSPEC, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
  • Sayfa Sayıları: ss.1171-1203
  • Anahtar Kelimeler: factional NLS, Fractional KdV, periodic waves, stability
  • Trakya Üniversitesi Adresli: Hayır

Özet

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 0$]]>, there is a travelling wave solution to fKdV and fNLS, 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.