Logarithmic Time Decay for the Cubic Nonlinear Schrödinger Equations

Nakao Hayashi, Pavel I. Naumkin

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)


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).

Original languageEnglish
Pages (from-to)5606-5643
Number of pages38
JournalInternational Mathematics Research Notices
Issue number14
Publication statusPublished - 2015
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)


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