Large time behavior for the cubic nonlinear Schrödinger equation

Nakao Hayashi, Pavel I. Naumkin

研究成果: Article査読

18 被引用数 (Scopus)

抄録

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.

本文言語English
ページ(範囲)1065-1085
ページ数21
ジャーナルCanadian Journal of Mathematics
54
5
DOI
出版ステータスPublished - 2002 10月
外部発表はい

ASJC Scopus subject areas

  • 数学 (全般)

フィンガープリント

「Large time behavior for the cubic nonlinear Schrödinger equation」の研究トピックを掘り下げます。これらがまとまってユニークなフィンガープリントを構成します。

引用スタイル