### Abstract

We continue to study the asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation {iu_{t} + u_{xx} + ia(|u|^{2}u)_{x} = 0, (t, x) ∈ R × R, (DNLS) {u(0, x) = u_{0}(x), x ∈ R, where a ∈ R. We prove that if ||u_{0}||_{H1,y} + ||u_{0}||_{H1+y,0} is sufficiently small with γ > 1/2, then the solution of (DNLS) satisfies the time decay estimate ||u(t)||_{L∞} + ||u_{x}(t)||_{L∞} ≤ C(1 + |t|)^{-1/2}, where H^{m,s} = {f ∈ S′; ||f||_{m,s} = ||(1 + |x|^{2})^{s/2}(1 - ∂^{2}_{x})^{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 language | English |
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Pages (from-to) | 383-400 |

Number of pages | 18 |

Journal | Discrete and Continuous Dynamical Systems |

Volume | 3 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1997 |

Externally published | Yes |

### ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics