两类奇异扰动型薛定谔方程组解的渐近分析

Schrödinger equations(systems) appear in different physical contexts, such as nonlinear optics，Bose-Einstein condensation and so on. In this project, we study the asymptotic behavior of solutions for two classes of singularly perturbed Schrödinger systems. One of them is the Schrödinger parabolic system only with nonlinear coupled term, for which we are going to adopt blow-up analysis technique and develop Almgren monotonicity formula and Alt-Caffarelli-Friedman monotonicity formula to study the uniform Lipschitz bound of solutions with respect to the nonlinear coupling constant. The other is the Schrödinger elliptic system with linear and nonlinear coupling terms(doubly coupled for short), for which we will investigate the asymptotic behavior of solutions as the nonlinear coupling constant goes to negative infinity. Moreover, we will study the regularity of the limiting profile, the characterization of the free boundary and the relationship between the limiting profile and the optimal partition for the principal eigenvalue of elliptic systems. In addition, as to doubly coupled Schrödinger system, we also intend to study how the existence of solutions depends on the parameters of equations. The study on the behaviors of solutions to Schrödinger systems can not only promote the development of the theory in nonlinear functional analysis, but also reveal some inherent laws in the physical area (especially in quantum mechanics).

薛定谔方程（组）来源于物理学中多个领域，如非线性光学、玻色-爱因斯坦凝聚等。本项目研究两类奇异扰动型薛定谔方程组解的渐近行为，一类是仅含非线性耦合项的抛物型薛定谔方程组，拟采用blow-up分析技术以及发展Almgren型和Alt-Caffarelli-Friedman型单调公式来研究方程组的解关于非线性耦合系数充分大时的一致Lipschitz正则性估计。另外一类是含有线性和非线性耦合项（简称双耦合）的椭圆型薛定谔方程组，拟研究方程组随着非线性耦合系数趋于负无穷时解的渐近行为以及产生的极限解的正则性、自由边界的刻画等，并且探究极限分布与一类椭圆方程组的主特征值区域最优分割之间的联系。此外，对于双耦合薛定谔系统，本项目还打算研究解的存在性与参数之间的关系。对于薛定谔方程组解的性态研究，不仅能促进非线性泛函分析这一领域理论的发展，更有利于揭示物理世界中(尤其是量子力学)的一些内在规律。

薛定谔方程(组)是数学研究中的热点前沿问题，在许多学科特别是量子力学中有着广泛的应用。本项目聚焦于薛定谔方程解的相关性态研究，运用变分方法、临界点理论等取得了如下重要结果：一是运用Almgren型和Alt-Caffarelli-Friedman型两类单调公式研究了强竞争型薛定谔方程组极限分布的性质，比如正则性、自由边界的刻画等，二是利用Pohozaev 流形和环绕定理的办法研究了分数阶薛定谔方程正规化解的存在性、多解性等，给出了这类方程解关于参数的精细刻画，三是利用集中紧性办法研究了伪相对论和双调和算子这两类方程驻波解的存在性和轨道稳定性。这些结果加深了我们对于薛定谔方程理论的理解，有助于阐释相关的物理现象。