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

Nakao Hayashi, Pavel I. Naumkin

13 被引用数 (Scopus)

## 抄録

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].

本文言語 English 383-400 18 Discrete and Continuous Dynamical Systems 3 3 https://doi.org/10.3934/dcds.1997.3.383 Published - 1997 はい

• 分析
• 離散数学と組合せ数学
• 応用数学

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