Asymptotic stability of small solitary waves for nonlinear Schrödinger equations with electromagnetic potential in ℝ³

2010/12/01 by Eva Koo, Koo, Eva
Mathematics · Physics and Astronomy · #35Bxx #35Pxx #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Spectral Theory in Mathematical Physics #math.AP #msc:35Bxx #msc:35Pxx

paper · pdf · doi:10.48550/arxiv.1012.0092

34 pages

arxiv created 2010/12/01 · openalex publication_date 2010/12/01 · arxiv updated 2010/12/02 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We consider the nonlinear magnetic Schrödinger equation for u: ℝ³ × ℝ → ℂ , iu_t = (i ∇ + A)² u + V u + g(u), u(x,0) = u₀(x), where A :ℝ³ → ℝ³ is the magnetic potential, V : ℝ³ → ℝ is the electric potential, and g = ± | u |² u is the nonlinear term. We show that under suitable assumptions on the electric and magnetic potentials, if the initial data is small enough in H¹ , then the solution of the above equation decomposes uniquely into a standing wave part, which converges as t → ∞ and a dispersive part, which scatters.

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