论文标题
非本地NLS方程的全球保守解决方案超出了爆炸
Global conservative solutions of the nonlocal NLS equation beyond blow-up
论文作者
论文摘要
我们考虑可集成的非局部非线性schrödinger（nnls）方程的库奇问题H^{1,1}（\ Mathbb {r}）$。众所周知，NNLS方程是可以集成的，并且具有孤子溶液，可以具有孤立的有限时间爆破点。这项工作的主要目的是提出一个合适的概念，以延续较弱的$ h^{1,1} $的库奇问题的本地解决方案（尤其是那些承认长期孤儿解决方案的解决方案）以外的奇异性。我们的主要工具是以Riemann-Hilbert问题形式的反向散射转换方法与非线性分散方程的PDE存在理论相结合。
We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation $ \I\partial_t q(x,t)+\partial_{x}^2q(x,t)+2σq^{2}(x,t)\overline{q(-x,t)}=0 $ with initial data $q(x,0)\in H^{1,1}(\mathbb{R})$. It is known that the NNLS equation is integrable and it has soliton solutions, which can have isolated finite time blow-up points. The main aim of this work is to propose a suitable concept for continuation of weak $H^{1,1}$ local solutions of the general Cauchy problem (particularly, those admitting long-time soliton resolution) beyond possible singularities. Our main tool is the inverse scattering transform method in the form of the Riemann-Hilbert problem combined with the PDE existence theory for nonlinear dispersive equations.
