在半经典极限下修正的非线性薛定谔方程：超音速、亚音速和跨音速特性
On the modified nonlinear Schrödinger equation in the semiclassical limit: supersonic, subsonic, and transsonic behavior
摘要：
在半经典极限下，比较了修正的非线性薛定谔（MNLS）方程与（未修正的）非线性薛定谔（NLS）方程的聚焦和散焦变形.我们描述方面的限制动力学和讨论如何从理论上通过逆散射和非交换最速下降法的动力学性质是显而易见的。主要的信息是，根据初始数据，MNLS方程可以表现得像散焦NLS方程，像聚焦NLS方程（在这两种情况下，当NLS方程被适当修改的初始数据时，类比在半经典极限中是渐近准确的），或者像两者的有趣混合。在后一种情况下，我们确定了一个功能的动态类似于气体动力学中的音速线，一个自由边界分离亚音速流从超音速流。
英文摘要：
The purpose of this paper is to present a comparison between the modified nonlinear Schrödinger (MNLS) equation and the focusing and defocusing variants of the (unmodified) nonlinear Schrödinger (NLS) equation in the semiclassical limit. We describe aspects of the limiting dynamics and discuss how the nature of the dynamics is evident theoretically through inverse-scattering and noncommutative steepest descent methods. The main message is that, depending on initial data, the MNLS equation can behave either like the defocusing NLS equation, like the focusing NLS equation (in both cases the analogy is asymptotically accurate in the semiclassical limit when the NLS equation is posed with appropriately modified initial data), or like an interesting mixture of the two. In the latter case, we identify a feature of the dynamics analogous to a sonic line in gas dynamics, a free boundary separating subsonic flow from supersonic flow.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
