非线性薛定谔方程孤立波解的相关问题研究

Nonlinear Schr?dinger equations (NLS) are the fundamental equations describing quantum mechanics and apply to many physical fields such as nonlinear optics, superconductivity and Bose-Einstein concentration (BEC) and so on. The study on solitary wave solutions, including travelling wave and standing wave solutions, plays an important role in both theory and applications of NLS. In this subject, we will focus on two classes of NLS: one is nonlinear Schr?dinger-Poisson equation from superconductivity, and the other is Gross-Pitaevskii equation from BEC. We will study the existence of nodal type standing wave solution of nonlinear Schr?dinger-Poisson equation by using variational methods and critical point theory.and try to find new methods to construct nodal solutions of nonlinear elliptic equations in unbounded domains with nonlocal terms. We will also study the stability of travelling wave solution of Gross-Pitaevskii equation and give some new ideas to determine the stability of solitary wave solutions of NLS. This subject will involve some new applications of variational methods and critical point theory in nonlinear functional analysis.

非线性薛定谔方程是描述量子力学的基本方程，它在非线性光学、超导、玻色-爱因斯坦凝聚等物理领域有着广泛的应用。非线性薛定谔方程孤立波解（包括行波解和驻波解）的研究有着重要的理论及应用意义。本项目将主要围绕两类重要的非线性薛定谔方程：一类是超导中的非线性Schr?dinger -Poisson方程，另一类是来自于玻色-爱因斯坦凝聚中的Gross-Pitaevskii方程, 对其孤立波解的存在性和稳定性开展深入的数学理论研究。具体研究内容包括：1. 利用变分法和临界点理论研究非线性Schr?dinger-Poisson方程变号驻波解的存在性, 探索无界域上含有非局部项的非线性椭圆型偏微分方程变号解的构造方法. 2. 研究Gross-Pitaevskii 方程行波解的稳定性，探索判别非线性薛定谔方程孤立波解稳定性的新方法。该项目的研究将涉及到非线性泛函分析中诸如变分法及临界点理论的新应用。

具有非零边界条件的Gross–Pitaevskii（GP）方程来源于非线性光学中暗孤子的研究。这类GP方程是一类特殊的反聚焦型非线性Schrodinger方程，空间无穷远处的非零边界条件使得解的结构比经典的零边界情形更为复杂。上世纪八十年代，英国物理学家Paul H. Roberts教授等对GP 方程的行波解进行了长期系统的数值研究，并且基于实验结果提出了一系列的猜测：包括行波解的存在性、稳定性、对称性等。这些关于行波解的猜测被统称为"Roberts programme"。从上世纪九十年代末开始，法国数学家Bethuel, Saut等对"Roberts programme"开展了严格的数学理论研究，并在高维行波解的存在性方面取得了重要的进展，但是关于高维行波解的稳定性没有任何结果。法国学者Maris曾在论文【Annals of Mathematics, 2013, Page 112】中指出“A much more difficult problem is to understand the stability of traveling waves...the orbital stability as well as the asymptotic stability of traveling waves in dimensions larger than two are still open problems”. 在本项目中，我们主要研究了GP 方程三维行波解的轨道稳定性问题，证明了：如果行波速度充分小，行波解在能量空间里一定是轨道稳定的；如果行波速度接近音速时，行波解在能量空间里一定是轨道不稳定的。这一研究结果发表在【Archive for Rational Mechanics and Analysis, vol. 222, 2016】上面，回答了Maris在论文【Annals of Mathematics, 2013, Page 112】中提出的关于三维行波解轨道稳定性的公开问题，从数学上严格验证了"Roberts programme"中关于GP 方程三维行波解的稳定性猜测，从方法上拓展了Grillakis-Shatah-Strauss经典稳定性理论的适用范围。另外，我们利用变分理论和拓扑度方法证明了Schrodinger-Poisson 方程变号解的存在性。