Self-similar blow-up profiles for slightly supercritical nonlinear\n Schr "odinger equations

2019/11/26 by Yakine Bahri, Bahri, Yakine, Yvan Martel +3 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1911.11457

openalex publication_date 2019/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct radially symmetric self-similar blow-up profiles for the mass\nsupercritical nonlinear Schr "odinger equation i\∂_t u + \Δ u +\n|u|^p-1u=0 on \R^d, close to the mass critical case and for any\nspace dimension d\≥ 1. These profiles bifurcate from the ground state\nsolitary wave. The argument relies on the classical matched asymptotics method\nsuggested in [Sulem, C.; Sulem, P.-L., The nonlinear Schr "odinger equation.\nSelf-focusing and wave collapse. Applied Mathematical Sciences, 139.\nSpringer-Verlag, New York, 1999] which needs to be applied in a degenerate case\ndue to the presence of exponentially small terms in the bifurcation equation\nrelated to the log-log blow-up law observed in the mass critical case.\n

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• Self-similar blowup for mass supercritical Schrödinger equations
• Mode stability for self-similar blowup of slightly supercritical NLS: II. high-energy spectrum

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