arXiv 2108.096772021-08math.APnlin.SI

具有非零背景的散焦NLS方程：无孤子区的大时间渐近性

The defocusing NLS equation with nonzero background: Large-time asymptotics in the solitonless region

Zhaoyu Wang, Engui Fan

备注：: 53 pages

摘要：

研究了具有非零背景$$\begin{align} &iq_t+q_{xx}-2(|q|^2-1)q=0, \nonumber\\ &q(x,0)=q_0(x), \quad \lim_{x \to \pm \infty}q_0(x)=\pm 1. \end{align}$$的散焦Schr $\ddot{\text{o}}$ dinger（NLS）方程的Cauchy问题最近，Cuccagna和Jenkins利用Deift-Zhou非线性最速下降法的$\bar{\partial}$推广，在时空区域$|x/(2t)|<1$（在跳跃轮廓上没有定相点的孤子区域）中，证明了NLS方程的$N$ -孤子解的渐近稳定性.它们的大时间渐近展开形式为\begin{align} q(x,t)= T(\infty)^{-2} q^{sol,N}(x,t) + \mathcal{O}(t^{-1 }),\label{res1} \end{align}，其首项为N孤子，第二项$\mathcal{O}(t^{-1})$为$\overline\partial$ -方程的残差.本文主要研究孤子区域外的时空区域$ |x/(2t)|>1$中的大时间渐近性，但在跳跃轮廓$\mathbb{R}$上会出现两个驻点。我们发现了一个不同于（\ref{res1}）其前导项为非零背景的$$\begin{align} q(x,t)= e^{-iα(\infty)} \left(1 +t^{-1/2} h(x,t) \right)+\mathcal{O}\left(t^{-3/4}\right),\label{res2} \end{align}$$，第二项$t^{-1/2}$来自连续谱，第三项$\mathcal{O}(t^{-3/4})$来自一个$\overline\partial$ -方程的残差，上述两个渐近结果（\ref{res1}）和（\ref{res2}）暗示了Cuccagna和Jenkins所考虑的区域$ |x/(2t)|<1$是一个快速衰减的孤子解区域，而我们所考虑的区域$ |x/(2t)|>1$是一个缓慢衰减的非零背景区域。

英文摘要：

We consider the Cauchy problem for the defocusing Schr$\ddot{\text{o}}$dinger (NLS) equation with a nonzero background $$\begin{align} &iq_t+q_{xx}-2(|q|^2-1)q=0, \nonumber\\ &q(x,0)=q_0(x), \quad \lim_{x \to \pm \infty}q_0(x)=\pm 1. \end{align}$$ 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 $\bar{\partial}$ generalization of the Deift-Zhou nonlinear steepest descent method. Their large-time asymptotic expansion takes the form \begin{align} q(x,t)= T(\infty)^{-2} q^{sol,N}(x,t) + \mathcal{O}(t^{-1 }),\label{res1} \end{align} whose leading term is N-soliton and the second term $\mathcal{O}(t^{-1})$ is a residual error from a $\overline\partial$-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 $\mathbb{R}$. We found a asymptotic expansion that is different from (\ref{res1}) $$\begin{align} q(x,t)= e^{-iα(\infty)} \left(1 +t^{-1/2} h(x,t) \right)+\mathcal{O}\left(t^{-3/4}\right),\label{res2} \end{align}$$ whose leading term is a nonzero background, the second $t^{-1/2}$ order term is from continuous spectrum and the third term $\mathcal{O}(t^{-3/4})$ is a residual error from a $\overline\partial$-equation.The above two asymptotic results (\ref{res1}) and (\ref{res2}) 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.

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