Global existence of small solutions to the generalized derivative nonlinear Schrödinger equation

We study the global in time existence of small solutions to the generalized derivative nonlinear Schrödinger equations of the form i∂_tu + (1/2)Δu = N(u, ∇u, ū ∇ū), (t, x) ∈ R × Rⁿ, (A) u(0, x) = u₀(x), x ∈ Rⁿ, where the space dimension n ≥ 3, the initial data u₀ are sufficiently small, ū is the complex conjugate of u and the nonlinear term N is a smooth complex valued function C × Cⁿ × C × Cⁿ → 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_αβγu^α1ū ^α2w^βw̄^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 u₀ ∈ H^[n/2]+5,0 ∩H^[/2]+3,2, where H^m,s = {φ ∈ L²; ∥φ∥_m,s = ∥(1 + x²)^s/2(1 - Δ)^m/2φ∥_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.