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Virial functional and dynamics for nonlinear Schrödinger equations of local interactions

  • Takafumi Akahori
  • Hiroaki Kikuchi
  • Takeshi Yamada
Article
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Abstract

Our aim is to verify that the functional in the virial identity classifies the dynamics for nonlinear Schrödinger equations of local interactions. In particular, we give a condition under that there exist stable ground states. Our proof of this stability result is based on the ideas in Colin (Ann Inst H Pincaré 23:753–764, 2006) and Shatah (Math Phys 91:313–327, 1983). However, we emphasize that our argument does not use the strict convexity of the \(\dot{H}^{1}\)-norm of ground state with respect to \(\omega \): a key lemma is Lemma 4.8 below. Furthermore, we discuss the limiting profile of ground states (see Theorem 4.4).

Keywords

Virial functional Ground state Stability Scattering Blowup Variational method Limiting profile 

Mathematics Subject Classification

35J20 35Q55 37K40 37K45 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Takafumi Akahori
    • 1
  • Hiroaki Kikuchi
    • 2
  • Takeshi Yamada
    • 3
  1. 1.Shizuoka UniversityHamamatsuJapan
  2. 2.Tsuda UniversityKodairaJapan
  3. 3.NagoyaJapan

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