Large Time Behavior of Solutions for Derivative Cubic Nonlinear Schrödinger Equations

Abstract

We study the asymptotic behavior in time and scattering problem for the solutions to the Cauchy problem for the derivative cubic nonlinear Schrodinger equations of the following form

(A)    iut+uxx = N(u, u, ux, u__x), t ∈ ℝ, x ∈ ℝ; u(0, x) = _u_0(x), x ∈ ℝ,

where

N(u, u, ux, u__x) = ℋ1|u|2 u + _i_ℋ2|u|2 ux + _i_ℋ3_u_2 u__x + ℋ4|ux|2 u + ℋ5_u__ux_2 + _i_ℋ6|ux|2 ux,

ℋ1 = ℋ1(|u|2), ℋ1(z) ∈ _C_3(ℝ+); ℋ1(z) = _λ_1 + O(z), as z → +0, ℋ1, ℋ6 are real valued functions. Here the parameters _λ_1, _λ_6 ∈ ℝ, and _λ_2, _λ_3, _λ_4, _λ_5 ∈ ℂ are such that _λ_2 – _λ_3 ∈ ℝ and _λ_4 – _λ_5 ∈ ℝ. If ℋ5(z) = _λ_5/(1+μz) and _λ_5 = μ = ∓ 1, ℋ1 = ℋ2 = ℋ3 = ℋ4 = ℋ5 = ℋ6 = 0 equation (A) appears in the classical pseudospin magnet model [9]. We prove that if _u_0 ∈ _H_3,0 ∩ _H_2,1 and the norm ‖_u_0‖3,0 + ‖_u_0‖2,1 = ε is sufficiently small, then the solution of (A) exists globally in time and satisfies the sharp time decay estimate ‖u(t)‖2,0, ∞ ≤ Cε(1+|t|)-1/2, where ‖φ‖m,s,p = ‖(1+x_2)s/2(1−∂_x_2)m/2_φ‖Lp, Hpm,s ={φ ∈ S'; ‖φ‖m,s,p < ∞}. Furthermore we prove existence of modified scattering states and nonexistence of nontrivial scattering states. Our method is based on a certain gauge transformation and an appropriate phase function.