Global dynamics above the first excited energy for the nonlinear Schrödinger equation with a potential

2016/03/08 by Kenji Nakanishi, Nakanishi, Kenji · 1 citation
Mathematics · Physics and Astronomy · #35Q55 #37D10 #37K40 #37K45 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Cold Atom Physics and Bose-Einstein Condensates #FOS: Mathematics #Nonlinear Photonic Systems

paper · pdf · doi:10.48550/arxiv.1603.02361

openalex publication_date 2016/03/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider the focusing nonlinear Schrödinger equation with a potential with a single negative eigenvalue. It has solitons with negative small energy, which are asymptotically stable, and solitons with positive large energy, which are unstable. We classify the global dynamics into 9 sets of solutions in the phase space including both solitons, restricted by small mass, radial symmetry, and an energy bound slightly above the second lowest one of solitons. The classification includes a stable set of solutions which start near the first excited solitons, approach the ground states locally in space for large time with large radiation to the spatial infinity, and blow up in negative finite time.

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• On the Spectrum of Schrödinger Operators Interacting at Two Distinct Scales

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