具有非零边界条件的导数薛定谔方程的长时间渐近性
Long time asymptotic behavior for the derivative Schrödinger equation with nonzero boundary conditions
摘要：
本文利用$\overline\partial$最速下降法研究了一类带非零边界条件的非线性薛定谔方程的Cauchy问题，其中$|q_\pm|=1$为\begin{align} &iq_{t}+q_{xx}+iσ(|q|^2q)_{x}=0,\\ & (x,0) = q_0(x), \quad\lim_{x\to\pm\infty} q_0(x) = q_\pm,\end{align}.基于拉克斯对的谱分析，我们将导数非线性薛定谔方程的解表示为Riemann-Hilbert问题的解，在固定的时空孤子区域$-3<x/t<-1$中，计算了解$q(x,t)$的长时间渐近展开式，这意味着孤子分辨猜想，并可以用一个参数由局域孤子和调制的$N(Λ)$ -孤子来表征。孤子相互作用作为一个移动通过该区域;剩余误差阶$\mathcal{O}( t^{-3/4})$从一个$\overline\partial$方程。
英文摘要：
In this paper, we apply $\overline\partial$ steepest descent method to study the Cauchy problem for the derivative nonlinear Schrödinger equation with nonzero boundary conditions \begin{align} &iq_{t}+q_{xx}+iσ(|q|^2q)_{x}=0,\\ & (x,0) = q_0(x), \quad\lim_{x\to\pm\infty} q_0(x) = q_\pm,\end{align} where $|q_\pm|=1$. Based on the spectral analysis of the Lax pair, we express the solution of the derivative nonlinear Schrödinger equation in terms of solutions of a Riemann-Hilbert problem.In a fixed space-time solitonic region $-3<x/t<-1$, we compute the long time asymptotic expansion of the solution $q(x,t)$,which implies soliton resolution conjecture and can be characterized with an $N(Λ)$-soliton whose parameters are modulated bya sum of localized soliton-soliton interactions as one moves through the region; the residual error order $\mathcal{O}( t^{-3/4})$ from a $\overline\partial$ equation.
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