Large time asymptotics of solutions to the periodic problem for the quadratic nonlinear Schrödinger equation

We study the large time asymptotics of solutions to the periodic problem for the quadratic nonlinear Schrödinger equation u t + i u xx = - u 2 , x ∈ - π , π , t > 0 , u ( 0 , x ) = ϕ x , x ∈ - π , π . We assume that the initial data ϕ x are 2 π - periodic and have small amplitude. Then we show th...

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Bibliographic Details
Published in:Nonlinear differential equations and applications 2023-03, Vol.30 (2), Article 23
Main Authors: Hayashi, Nakao, Naumkin, Pavel I.
Format: Article
Language:English
Subjects:
Analysis
Asymptotic properties
Mathematics
Mathematics and Statistics
Schrodinger equation
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Summary:We study the large time asymptotics of solutions to the periodic problem for the quadratic nonlinear Schrödinger equation u t + i u xx = - u 2 , x ∈ - π , π , t > 0 , u ( 0 , x ) = ϕ x , x ∈ - π , π . We assume that the initial data ϕ x are 2 π - periodic and have small amplitude. Then we show that the periodic solutions satisfy the following asymptotics u t - ε 1 + ε t L ∞ - π , π = O 1 + ε t - 2 as t → ∞ .
ISSN:1021-9722
1420-9004
DOI:10.1007/s00030-022-00830-y