Large time behavior of small solutions to subcritical derivative nonlinear schrödinger equations

Nakao Hayashi, Pavel I. Naumkin, Yasuko Yamazaki

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1 Citation (Scopus)

Abstract

We consider the derivative nonlinear Schrödinger equations { iut + 1/2uxx = a(t)F(u,ux, (t,x) ∈ R2, u(0,x) = εu0(x), x ∈ R, where the coefficient a (t) satisfies the time growth condition |a(t)| ≤ C (1+|t|)1-δ, 0 < δ < 1, ε is a sufficiently small constant and the nonlinear interaction term F consists of cubic nonlinearities of derivative type F(u, ux) = λ1 |u|2 u + iλ2 |u|2 ux + iλ3u2ūx + λ4 |ux|2u + λ5ūux2+iλ6|u x|2ux, where λ1, λ6 ∈ R, λ2, λ3, λ4, λ5 ∈ C, λ2 - λ3 7isin; R, and λ4 - λ5 ∈ R. We suppose that the initial data satifsfy the exponential decay conditions. Then we prove the sharp decay estimate ∥u(t)∥Lp, ≤ Cεt1/p-1/2, for all t ≥ 1, where 2 ≤ p ≤ ∞. Furthermore we show that for 1/2 < δ < 1 there exist the usual scattering states, when b(x) = λ1 - (λ2 - λ3) x + (λ4 - λ5) x2 - λ6x3 = 0, and the modified scattering states, when b(x) ≠ 0.

Original languageEnglish
Pages (from-to)779-789
Number of pages11
JournalProceedings of the American Mathematical Society
Volume130
Issue number3
DOIs
Publication statusPublished - 2002
Externally publishedYes

Keywords

  • Large time asymptotics
  • Scattering problem
  • Subcritical nonlinear Schrödinger equations

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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