Defocusing NLS equation with nonzero background: Large-time asymptotics in a solitonless region

We consider the Cauchy problem for defocusing Schrödinger (NLS) equation with a nonzero backgroundiqt+qxx−2(|q|2−1)q=0,q(x,0)=q0(x),limx→±∞⁡q0(x)=±1. Recently, space-time region |x/(2t)|<1 which is solitonic without stationary phase points on jump contour, Cuccagna and Jenkins presented asymptotic stability of N-soliton solutions NLS by using ∂¯ generalization Deift-Zhou nonlinear steepest descent method. Their large-time expansion takes form(0.1)q(x,t)=T(∞)−2qsol,N(x,t)+O(t−1), whose leading term second O(t−1) residual error from ∂‾-equation. In this paper, we are interested in asymptotics |x/(2t)|>1 outside soliton region, but there will be two appearing contour R. found an that different (0.1)(0.2)q(x,t)=e−iα(∞)(1+t−1/2h(x,t))+O(t−3/4), background, t−1/2 order continuous spectrum third O(t−3/4) The above results (0.1) (0.2) imply considered fast decaying solution while us slow background region.