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Asymptotic behavior for the long-range nonlinear Schrödinger equation on the star graph with the Kirchhoff boundary condition

Kazuki Aoki, Takahisa Inui, Hayato Miyazaki, Haruya Mizutani and Kota Uriya

Vol. 4 (2022), No. 2, 287–311
DOI: 10.2140/paa.2022.4.287
Abstract

We consider the cubic nonlinear Schrödinger equation on the star graph with the Kirchhoff boundary condition. We prove modified scattering for the final state problem and the initial value problem. Moreover, we also consider the failure of scattering for the Schrödinger equation with power-type long-range nonlinearities. These results are extensions of the results for the nonlinear Schrödinger equation on the one-dimensional Euclidean space.

Keywords
Schrödinger equation, star graph, long-range nonlinearity, modified scattering, failure of scattering
Mathematical Subject Classification
Primary: 35B40, 35Q55
Milestones
Received: 31 March 2021
Accepted: 11 January 2022
Published: 16 October 2022
Authors
Kazuki Aoki
Department of Mathematics, Graduate School of Science
Osaka University
Toyonaka-shi
Japan
Takahisa Inui
Department of Mathematics, Graduate School of Science
Osaka University
Toyonaka-shi
Japan
Department of Mathematics
University of British Columbia
Vancouver
Canada
Hayato Miyazaki
Teacher Training Courses, Faculty of Education
Kagawa University
Takamatsu-shi
Japan
Haruya Mizutani
Department of Mathematics, Graduate School of Science
Osaka University
Toyonaka-shi
Japan
Kota Uriya
Department of Applied Mathematics, Faculty of Science
Okayama University of Science
Okayama-shi
Japan