On long time asymptotic behavior of the defocusing schrodinger equation with finite density initial data

Zhaoyu Wang, Engui Fan

Published 2021-08-22Version 1

We consider the Cauchy problem for the defocusing Schr$\ddot{\text{o}}$dinger (NLS) equation with finite density initial data \begin{align} &iq_t+q_{xx}-2(|q|^2-1)q=0, \nonumber\\ &q(x,0)=q_0(x), \quad \lim_{x \to \pm \infty}q_0(x)=\pm 1.\nonumber\ \end{align} Recently for the space-time region $|x/t|<2$ without stationary points on the jump contour, Cuccagna and Jenkins presented the asymptotic stability of the $N$-soliton solutions for the NLS equation by using the $\bar{\partial}$ generalization of the nonlinear steepest descent method. Their asymptotic result is the form $$ q(x,t)= T(\infty)^{-2} q^{sol,N}(x,t) + \mathcal{O}(t^{-1 }).$$ The leading order term is $N$-soliton solutions and an error $\mathcal{O}(t^{-1})$. However, for the space-time region $|x/t|>2$ with two stationary points on the jump contour, whose long time asymptotics are still unknown and will be considered in this paper. We found the following asymptotic expansion $$ q(x,t)= e^{-i\alpha(\infty)} \left( q_{sol}(x,t;\sigma_d^{(out)}) +t^{-1/2} h(x,t) \right)+\mathcal{O}\left(t^{-3/4}\right).$$ which is different from Cuccagna-Jenkins's. The leading order term is $N$-soliton solutions; the second $t^{-1/2}$ order term is soliton-soliton and soliton-radiation interactions and the third term $\mathcal{O}(t^{-3/4})$ is a residual error of order from a $\overline\partial$ equation. Additionally, the asymptotic stability property for the N-soliton solutions of the defocusing NLS equation sufficiently is obtained.