Smoothing effects for some derivative nonlinear schrödinger equations

In this paper we study a smoothing property of solutions to the Cauchy problem for the nonlinear Schrödinger equations of derivative type: iu_t + u_xx = N(u, ū, u_x, ū_x), t ∈ R, x ∈ R; u(0, x) = u₀(x), x ∈ R, (A) where N(u, ū, u_x, ū_x) = K₁\u\²u + K₂\u\²u_x + K₃u²ū_x + K₄\u_x\²u + K₅ūu²_x + K₆\u_x\²u_x, the functions K_j = K_j(\u\²), K_j(z) ∈ C^∞([0,∞)). If the nonlinear terms N = ūu²_x/1+|u|² then equation (A) appears in the classical pseudospin magnet model [16]. Our purpose in this paper is to consider the case when the nonlinearity N depends both on u_x and ū_x. We prove that if the initial data u₀ ∈ H^3,∞ and the norms ||u₀||3,l are sufficiently small for any l ∈ N, (when N depends on ū_x), then for some time T > 0 there exists a unique solution u ∈ C^∞([-T,T]\{0};C^∞(R)) of the Cauchy problem (A). Here H^m,s = {φ ∈ L²; ||φ||_m,s < ∞}, ||φ||_m,s = ||(1 + x²)^s/2(1 - ∂²_x)^m/2φ||_L2, H^m,∞ = ∩_s≥1H^m,s.