论文标题
对于非零边界条件的导数Schrödinger方程的长期渐近行为
Long time asymptotic behavior for the derivative Schrödinger equation with nonzero boundary conditions
论文作者
论文摘要
在本文中，我们将$ \ operline \ partial $陡峭下降方法用于研究衍生化的非线性schrödinger方程，具有非零边界条件\ begin \ begin {align}＆iq_ {t}+q_ {t}+q_ {xx}+ix \ quad \ lim_ {x \ to \ pm \ pm \ infty} q_0（x）= q_ \ pm，\ end {align}其中$ | q_ \ pm | = 1 $。基于LAX对的频谱分析，我们根据Riemann-Hilbert问题的解决方案来表达衍生性非线性Schrödinger方程的解$ n（λ）$ - 索利顿的参数被调制为局部孤子 - 索里顿相互作用的总和，因为一个人通过该区域移动；从$ \ overline \ partial $方程中的剩余错误订单$ \ MATHCAL {O}（t^{ - 3/4}）$。
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.
