Soliton resolution for nonlinear Schrödinger type equations in the radial case

2023/04/09 by Avy Soffer, Xiaoxu Wu, Soffer, Avy +1 · 1 citation
Mathematics · Physics and Astronomy · #35Q55 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Photonic Systems

paper · pdf · doi:10.48550/arxiv.2304.04245

openalex publication_date 2023/04/09 · openalex created_date 2023/04/12 · openalex updated_date 2026/07/28

Abstract

We consider the Schrödinger equation with a general interaction term, which is localized in space, for radially symmetric initial data in n dimensions, n≥5. The interaction term may be space-time dependent and nonlinear. Assuming that the solution is bounded in H¹(ℝⁿ) uniformly in time, we prove soliton resolution conjecture (Asymptotic completeness) by demonstrating that the solution resolves into a smooth and localized part and a free radiation in L²_x(ℝⁿ) norm. Examples of such equations include inter-critical and super-critical nonlinear Schrödinger equations, saturated nonlinearities, time-dependent potentials, and combinations of these terms.

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