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Nonflat conformal blow-up profiles for the 1-dimensional critical nonlinear Schrödinger equation

Yvan Martel and Ivan Naumkin

Vol. 5 (2023), No. 3, 505–572

For the critical one-dimensional nonlinear Schrödinger equation, we construct blow-up solutions that concentrate a soliton at the origin at the conformal blow-up rate, with a nonflat blow-up profile. More precisely, we obtain a blow-up profile that equals |x| + iκx2 near the origin, where κ is a universal real constant. Such profile differs from the flat profiles obtained in the same context by Bourgain and Wang in 1997.

nonlinear Schrödinger equation, blow-up, soliton, critical equation
Mathematical Subject Classification
Primary: 35B44, 35Q41
Secondary: 37K40
Received: 30 November 2022
Accepted: 21 May 2023
Published: 2 November 2023
Yvan Martel
Laboratoire de mathématiques de Versailles
Université Paris-Saclay
Ivan Naumkin
Departamento de Física Matemática
Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas
Universidad Nacional Autónoma de México
Mexico City