Smoothing effects for some derivative nonlinear Schrödinger equations without smallness condition

Source abstract

In this paper we study a smoothing property of solutions to the Cauchy problem for the nonlinear Schrödinger equations of type : A iut+uxx=N(u,u¯,uxux¯), t∈ℝ,x∈ℝ;u(x,0)=u0(x), x∈ℝ, where N(u,u¯,ux,ux¯)=K1|u|2u+K2|u|2ux+K3u2ux¯+K4|ux|2u+K5u¯ux2+K6|ux|2ux, the functions Kj=Kj(|u|2) satisfy Kj∈C∞([0,+∞);ℂ). This equation has been derived from physics. For example if the nonlinear term is N(u)=u¯ux21+|u|2 then equation (A) appears in the classical pseudospin magnet model [18]. The aim of this paper is to study the case : the nonlinearity depends on ux and ux¯, and satisfies the so called Gauge condition : N(eiθu)=eiθN(u). We prove that if the initial data u0∈ℋ3,l for any l∈ℕ, then there exists a positive time T> 0 and a unique solution u∈C∞([−T,T]\{0};C∞(ℝ)) of the Cauchy problem (A). The result in this paper improves the previous one in [11] because we do not assume any size restriction on the data. Here ℋm,s={φ∈ℒ2(ℝ);‖φ‖m,s<+∞} and ‖ φ ‖m,s=‖ (1+x2)s/2(1−∂x2)m/2φ ‖ℒ2(ℝ),ℋm,∞=∩s≥1ℋm,s.