## Abstract

We study the global in time existence of small solutions to the generalized derivative nonlinear Schrödinger equations of the form i∂_{t}u + (1/2)Δu = N(u, ∇u, ū ∇ū), (t, x) ∈ R × R^{n}, (A) u(0, x) = u_{0}(x), x ∈ R^{n}, where the space dimension n ≥ 3, the initial data u_{0} are sufficiently small, ū is the complex conjugate of u and the nonlinear term N is a smooth complex valued function C × C^{n} × C × C^{n} → C. We assume that N is a quadratic function in the neighborhood of the origin and always includes at least one derivative, that is, |N(u, w, ū w̄)| ≤ C|w|(|u| + |w|), for small u and w in the case of space dimensions n = 3, 4. As a typical example we consider the case of the polynomial type nonlinearity of the form N(u, w, ū, w̄) = ∑ λ_{αβγ}u^{α1}ū ^{α2}w^{β}w̄^{γ} 2 ≤ |α| + |β| + |γ| ≤ l m ≤ |β| + |γ| ≤ l with w = (w_{j})_{1≤j≤n,} λ_{αβγ} ∈ C, l ≥ 2, m ≥ 1 for n = 3, 4, and m ≥ 0 for n ≥ 5. We prove the global existence of solutions to the Cauchy problem (A) under the condition that the initial data u_{0} ∈ H^{[n/2]+5,0} ∩H^{[/2]+3,2,} where H^{m,s} = {φ ∈ L^{2}; ∥φ∥_{m,s} = ∥(1 + x^{2})^{s/2}(1 - Δ)^{m/2}φ∥_{L2} < ∞} is the weighted Sobolev space. We also show the existence of the usual scattering states. Our result for n = 3, 4 is an improvement of Hayashi and Hirata, Nonlinear Anal. 31 (1998), 671-685.

Original language | English |
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Pages (from-to) | 133-147 |

Number of pages | 15 |

Journal | Asymptotic Analysis |

Volume | 21 |

Issue number | 2 |

Publication status | Published - 1999 Oct 1 |

Externally published | Yes |

## Keywords

- Derivative nonlinear Schrödinger equations
- General space dimensions
- Global small solutions

## ASJC Scopus subject areas

- Mathematics(all)