Large time behavior of small solutions to subcritical derivative nonlinear schrödinger equations

We consider the derivative nonlinear Schrödinger equations { iu_t + 1/2u_xx = a(t)F(u,u_x, (t,x) ∈ R², u(0,x) = εu₀(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) = λ₁ |u|² u + iλ₂ |u|² u_x + iλ₃u²ū_x + λ₄ |u_x|²u + λ₅ūu_x²+iλ₆|u _x|²u_x, where λ₁, λ6 ∈ R, λ₂, λ₃, λ₄, λ₅ ∈ C, λ₂ - λ₃ 7isin; R, and λ₄ - λ₅ ∈ 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) = λ₁ - (λ₂ - λ₃) x + (λ₄ - λ₅) x² - λ₆x³ = 0, and the modified scattering states, when b(x) ≠ 0.