Asymptotic behavior in time of solutions to the derivative nonlinear schrödinger equation revisited

Nakao Hayashi, Pavel I. Naumkin

Research output: Contribution to journalArticle

12 Citations (Scopus)

Abstract

We continue to study the asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation {iut + uxx + ia(|u|2u)x = 0, (t, x) ∈ R × R, (DNLS) {u(0, x) = u0(x), x ∈ R, where a ∈ R. We prove that if ||u0||H1,y + ||u0||H1+y,0 is sufficiently small with γ > 1/2, then the solution of (DNLS) satisfies the time decay estimate ||u(t)||L∞ + ||ux(t)||L∞ ≤ C(1 + |t|)-1/2, where Hm,s = {f ∈ S′; ||f||m,s = ||(1 + |x|2)s/2(1 - ∂2x)m/2 f||L2 < ∞}, m, s ∈ R. In the previous paper [4,Theorem 1.1] we showed the same result under the condition that γ ≥ 2. Furthermore we show the asymptotic behavior in time of solutions involving the previous result [4,Theorem 1.2].

Original languageEnglish
Pages (from-to)383-400
Number of pages18
JournalDiscrete and Continuous Dynamical Systems
Volume3
Issue number3
DOIs
Publication statusPublished - 1997
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Fingerprint Dive into the research topics of 'Asymptotic behavior in time of solutions to the derivative nonlinear schrödinger equation revisited'. Together they form a unique fingerprint.

  • Cite this