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

Abstract

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