Scattering problem and asymptotics for a relativistic nonlinear Schrödinger equation

We study the global existence and asymptotic behaviour in time of solutions of the Cauchy problem for the relativistic nonlinear Schrödinger equation in one space dimension iu_t + 1/2u_xx + script N = 0, (t, x) ∈ ℝ × ℝ; u(0, x) = u₀(x), x ∈ ℝ, (A) where script N = λ|u|²u + uf (|u|²) - ug′(|u|²)(g(|u|²))_xx, λ ∈ ℝ, the real-valued functions f and g are such that |f^(j)(z)| ≤ Cz^1+σ-j, j = 0, 1, 2, 3, for z → +0, where σ > 0, and g ∈ C⁵ ([0, ∞)). Equation (A) models the self-channelling of a high-power, ultra-short laser in matter if f(z) = 2λ(1 - z/2 - (1 + z)^-1/2, g(z) = √1 + z, for all z ≥ 0. When λ = 0, f = 0 equation (A) also has some applications in condensed matter theory, plasma physics, Heisenberg ferromagnets and fluid mechanics. We prove that if the norm of the initial data ∥u₀∥ _H3.0 + ∥u₀∥ _H0.3 is sufficiently small, where H^m,s = {φ ∈ S′; ∥φ∥_m,s = ∥(1+x²)^s/2(1 - ∂²_x)^m/2φ∥ _L2 < ∞}, then the solution of the Cauchy problem (A) exists globally in time and satisfies the sharp L^∞ time-decay estimate ∥u(t)∥ _L∞ ≤ C (1 + |t|)^-1/2. Furthermore, we prove the existence of the modified scattering states and the nonexistence of the usual scattering states by introducing a certain phase function when λ ≠ 0. On the other hand, the existence of the usual scattering states when λ. = 0 follows easily from our results.