## Abstract

We consider the derivative nonlinear Schrödinger equations { iu_{t} + 1/2u_{xx} = a(t)F(u,u_{x}, (t,x) ∈ R^{2}, u(0,x) = εu_{0}(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, u_{x}) = λ_{1} |u|^{2} u + iλ_{2} |u|^{2} u_{x} + iλ_{3}u^{2}ū_{x} + λ_{4} |u_{x}|^{2}u + λ_{5}ūu_{x}^{2}+iλ_{6}|u _{x}|^{2}u_{x}, where λ_{1}, λ6 ∈ R, λ_{2}, λ_{3}, λ_{4}, λ_{5} ∈ C, λ_{2} - λ_{3} 7isin; R, and λ_{4} - λ_{5} ∈ R. We suppose that the initial data satifsfy the exponential decay conditions. Then we prove the sharp decay estimate ∥u(t)∥_{Lp}, ≤ C_{εt1/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) = λ_{1} - (λ_{2} - λ_{3}) x + (λ_{4} - λ_{5}) x^{2} - λ_{6}x^{3} = 0, and the modified scattering states, when b(x) ≠ 0.

Original language | English |
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Pages (from-to) | 779-789 |

Number of pages | 11 |

Journal | Proceedings of the American Mathematical Society |

Volume | 130 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2002 |

## Keywords

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