ON THE STABILITY OF THE COMPACTON WAVES FOR THE DEGENERATE KDV AND NLS MODELS

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HAKKAEV S. A., Ramadan A., Stefanov A. G.

Quarterly of Applied Mathematics, cilt.80, sa.3, ss.507-528, 2022 (SCI-Expanded, Scopus) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 80 Sayı: 3
  • Basım Tarihi: 2022
  • Doi Numarası: 10.1090/qam/1616
  • Dergi Adı: Quarterly of Applied Mathematics
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Applied Science & Technology Source, Compendex, Computer & Applied Sciences, MathSciNet, zbMATH
  • Sayfa Sayıları: ss.507-528
  • Açık Arşiv Koleksiyonu: AVESİS Açık Erişim Koleksiyonu
  • Trakya Üniversitesi Adresli: Evet

Özet

In this paper, we consider the degenerate semi-linear Schrödinger and Korteweg-de Vries equations in one spatial dimension. We construct special solutions of the two models, namely standing wave solutions of NLS and traveling waves, which turn out to have compact support, compactons. We show that the compactons are unique bellshaped solutions of the corresponding PDEs and for appropriate variational problems as well. We provide a complete spectral characterization of such waves, for all values of p. Namely, we show that all waves are spectrally stable for 2 < p ≤ 8, while a single mode instability occurs for p > 8. This extends previous work of Germain, Harrop-Griffiths and Marzuola [Quart. Appl. Math. 78 (2020), pp. 1–32] who have previously established orbital stability for some specific waves, in the range p < 8