论文标题
在线上存在非局部Schrödinger方程的全球解决方案
Existence of global solutions to the nonlocal Schrödinger equation on the line
论文作者
论文摘要
在本文中，我们解决了与初始数据$ q_0（x）\ in H^{1,1}（\ r）$一起使用$ l^1（\ r^1（\ r^r）$ small-norm假设，$ q_0（x）\ in H^{1,1}中的最初数据$ q_0（x）\ in初始数据$ q_0（x）\ in the $ q_0（x）\ in the $ q_0（x）\ in the $ l^1（\ r）$ norm假设。我们严格地表明，非本地NLS方程的光谱问题不承认特征值或共振，以及周的消失引理在$ l^1（\ r）$小型假设下是有效的。 With inverse scattering theory and the Riemann-Hilbert approach, we rigorously establish the bijectivity and Lipschitz continuous of the direct and inverse scattering map from the initial data to reflection coefficients.By using reconstruction formula and the Plemelj projection estimates of reflection coefficients,we further obtain the existence of the local solution and the priori estimates, which assure the existence of the global solution to the Cauchy problem for the非本地NLS方程。
In this paper, we address the existence of global solutions to the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (nonlocal NLS) equation with the initial data $q_0(x)\in H^{1,1}(\R)$ with the $L^1(\R)$ small-norm assumption. We rigorously show that the spectral problem for the nonlocal NLS equation admits no eigenvalues or resonances, as well as Zhou vanishing lemma is effective under the $L^1(\R)$ small-norm assumption. With inverse scattering theory and the Riemann-Hilbert approach, we rigorously establish the bijectivity and Lipschitz continuous of the direct and inverse scattering map from the initial data to reflection coefficients.By using reconstruction formula and the Plemelj projection estimates of reflection coefficients,we further obtain the existence of the local solution and the priori estimates, which assure the existence of the global solution to the Cauchy problem for the nonlocal NLS equation.
