TY - JOUR

T1 - Logarithmic Time Decay for the Cubic Nonlinear Schrödinger Equations

AU - Hayashi, Nakao

AU - Naumkin, Pavel I.

N1 - Publisher Copyright:
© 2014 The Author(s) 2014. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com.
Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

PY - 2015

Y1 - 2015

N2 - We prove global existence of solutions to the Cauchy problem for the 1D cubic nonlinear Schrödinger equation iut + 1/2 uxx =λ eiπ/2 u3 + |u|2u, x ε R, t>1, (0.1) where λ ε R, 0<|λ|< √ 3.We show that the time decay estimate of the solution in the far region |x| > √ t coincides with that for the linear case, whereas in the short-range region |x| ≤ √ t the solution obtains an extra logarithmic time decay, which is less than that in the absence of the resonance term |u|2uin Equation (0.1).

AB - We prove global existence of solutions to the Cauchy problem for the 1D cubic nonlinear Schrödinger equation iut + 1/2 uxx =λ eiπ/2 u3 + |u|2u, x ε R, t>1, (0.1) where λ ε R, 0<|λ|< √ 3.We show that the time decay estimate of the solution in the far region |x| > √ t coincides with that for the linear case, whereas in the short-range region |x| ≤ √ t the solution obtains an extra logarithmic time decay, which is less than that in the absence of the resonance term |u|2uin Equation (0.1).

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U2 - 10.1093/imrn/rnu102

DO - 10.1093/imrn/rnu102

M3 - Article

AN - SCOPUS:84939640517

VL - 2015

SP - 5606

EP - 5643

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 14

ER -