Large time behavior for the cubic nonlinear Schrödinger equation

Nakao Hayashi, Pavel I. Naumkin

Research output: Contribution to journalArticle

16 Citations (Scopus)

Abstract

We consider the Cauchy problem for the cubic nonlinear Schrödinger equation in one space dimension (1) {iut + 1/2uxx + u-3 = 0, t ∈ R, x ∈ R, u(0, x) = u0(x), x ∈ R.} Cubic type nonlinearities in one space dimension heuristically appear to be critical for large time. We study the global existence and large time asymptotic behavior of solutions to the Cauchy problem (1). We prove that if the initial data u0 ∈ H1,0 ∩ H0,1 are small and such that sup|ξ| ≤ 1 | arg ℱu0(ξ) - πn/2 | < π/8 for some n ∈ Z, and inf|ξ| ≤ 1 |ℱu0(ξ)| > 0, then the solution has an additional logarithmic time-decay in the short range region |x| ≤ √t. In the far region |x| > √t the asymptotics have a quasilinear character.

Original languageEnglish
Pages (from-to)1065-1085
Number of pages21
JournalCanadian Journal of Mathematics
Volume54
Issue number5
DOIs
Publication statusPublished - 2002 Oct
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)

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