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On Asymptotic Stability in Energy Space of Ground States for Nonlinear Schrödinger Equations

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Abstract

We consider nonlinear Schrödinger equations

$$iu_t +\Delta u +\beta (|u|^2)u=0\, ,\, \text{for} (t,x)\in \mathbb{R}\times \mathbb{R}^d,$$

where d ≥ 3 and β is smooth. We prove that symmetric finite energy solutions close to orbitally stable ground states converge to a sum of a ground state and a dispersive wave as t → ∞ assuming the so called the Fermi Golden Rule (FGR) hypothesis. We improve the “sign condition” required in a recent paper by Gang Zhou and I.M.Sigal.

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Authors and Affiliations

  1. DISMI University of Modena and Reggio Emilia, via Amendola 2, Padiglione Morselli, Reggio Emilia, 42100, Italy

    Scipio Cuccagna

  2. Faculty of Mathematics, Kyushu University, 6-10-1, Hakozaki, Higashi-ku, Fukuoka, 812-8581, Japan

    Tetsu Mizumachi

Authors
  1. Scipio Cuccagna
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  2. Tetsu Mizumachi
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Correspondence to Scipio Cuccagna.

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Communicated by H.-T. Yau

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Cuccagna, S., Mizumachi, T. On Asymptotic Stability in Energy Space of Ground States for Nonlinear Schrödinger Equations. Commun. Math. Phys. 284, 51–77 (2008). https://doi.org/10.1007/s00220-008-0605-3

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  • DOI: https://doi.org/10.1007/s00220-008-0605-3

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