Failure of scattering to standing waves for a Schrödinger equation with long-range nonlinearity on star graph

We consider the Schrödinger equation with power type long-range nonlinearity on star graph. Under a general boundary condition at the vertex, including Kirchhoff, Dirichlet, δ , or δ ′ boundary condition, we show that the non-trivial global solution does not scatter to standing waves. Our proof is b...

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Bibliographic Details
Published in:Journal of evolution equations 2021-03, Vol.21 (1), p.297-312
Main Authors: Aoki, Kazuki, Inui, Takahisa, Mizutani, Haruya
Format: Article
Language:English
Subjects:
Analysis
Boundary conditions
Dirichlet problem
Euclidean geometry
Euclidean space
Mathematics
Mathematics and Statistics
Nonlinearity
Scattering
Schrodinger equation
Solitary waves
Standing waves
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Summary:We consider the Schrödinger equation with power type long-range nonlinearity on star graph. Under a general boundary condition at the vertex, including Kirchhoff, Dirichlet, δ , or δ ′ boundary condition, we show that the non-trivial global solution does not scatter to standing waves. Our proof is based on the argument by Murphy and Nakanishi (Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations), who treated the long-range nonlinear Schrödinger equation with a general potential in the Euclidean space, in order to consider general boundary conditions.
ISSN:1424-3199
1424-3202
DOI:10.1007/s00028-020-00579-w