非线性Schrödinger方程解的爆破图景分析与驻波的条件稳定性

For nonlocal nonlinear Schrödinger equations, quasilinear Schrödinger equations and inhomogeneous nonlinear Schrödinger equation, proper functionals and variational problems are constructed to derive the variational characterization of the ground standing waves and its connection with the evolution equations. Using the variational characterization, singular part is constructed in terms of physical properties and numerical results on the singular structure of blow-up solutions. The blow-up solutions are decomposed into a singular part and a remainder term. The equation for the remainder term is established. The regularity and long time behavior of the remainder are analyzed, and the limiting profile of blow-up solutions is established. At the same time, the invariant sets are constructed in terms of the variational characterization, and the blow-up threshold obtained. As a result, the instability of standing waves is achieved. The essential characteristic of blow-up and instability mechanism of standing wave are explored from the profile and threshold condition of blow-up solution. On this basis, the spectrum structure of the linearized Schrödinger operators and corresponding stable/unstable subspace is studied. Moreover, some tools such as operator semigroup and fixed point theorem are used to investigate the conditional stability of standing waves, that is, the stability of the standing wave at given directions or manifolds.

针对非局部非线性Schrödinger方程、拟线性Schrödinger方程及非齐次的非线性Schrödinger方程，构造合适的泛函与约束变分问题，挖掘基态驻波解的变分特征及其与系统发展流之间的联系。对照其物理性质及关于奇性结构的数值结论，利用基态的变分特征构造爆破解的奇性部分，将爆破解分解为奇性部分与余项之和。然后建立余项满足的方程式并研究余项的正则 性与渐近行为，从而分析爆破解的极限图景。同时，依托驻波的变分特征构造不变集，获取爆 破解的门槛条件，进而研究驻波的不稳定性。然后，透过爆破图景与爆破门槛条件探究爆破的本质特征及驻波不稳定的机制。在此基础上，依靠驻波处线性化Schrödinger算子的谱结构及由此形成的线性化方程的稳定性与不稳定子空间，运用算子半群、不动点等工具探究驻波的条件稳定性，即驻波在特定方向或特定流形的稳定性。

非线性波在水波、非线性光学、玻色爱因斯坦凝聚等物理及应用科学中随处可见。爆破、孤波的存在性与稳定性、散射是非线性波的重要研究主题。非线性Schrödinger方程是研究这些主题的简单但非平凡数学模型。本项目研究非线性Schrödinger方程爆破解的极限图景与驻波解的条件稳定性。对三维空间的广义Davey-Stewartson系统我们给出其爆破解的极限图景，我们的方法可适用于非质量临界且不具有尺度不变性的非线性Schrödinger方程。对具逆平方势的非线性Schrödinger方程我们建立了驻波的轨道稳定的判别准则，并对质量临界的强不稳定驻波得到了条件稳定性。此外我们还研究了Zakharov-Kuznetsov方程的不稳定孤波，通过不变流形刻画起孤波的条件稳定性。