非线性Schrödinger方程的整体动力学

In the recent studies of macroscopic quantum phenomena, when the external trapping potential is restricted on the partial directions, the nonlinear Schrödinger equations with partial confinements are proposed as a new model. In this project, based on studies of the two classical nonlinear Schrödinger equations without confinement and with harmonic confinement, we develop new variation methods to investigate the global dynamics for the new model. In terms of the invariant sets for evolution system, we research the sharp thresholds of existence of global solutions for the new model. We search the relationships among differential thresholds, and the optimal invariant sets for the existence of global solutions. We determine the evolution profile and asymptotic behavior of the global solutions in different invariant sets. And we further investigate the existence of standing waves with different frequencies and their stabilities. Based on Tao’s conjecture of soliton resolution, we study standing wave resolution of global solutions for the above nonlinear Schrödinger equations. By using the results of the new model and the relations among three classes of nonlinear Schrödinger equations, we promote the studies of the classical nonlinear Schrödinger equations, and reveal the dynamical profiles between global solutions and standing waves. This will give impetus to global dynamics for nonlinear Schrödinger equations. Furthermore, according to physical experiments and numerical computations for the new model, we realize the mathematical expressions and dynamical predictions for some new quantum phenomena.

在宏观量子现象的最近研究中，当外势限制在部分方向上时，提出了带部分限制的非线性Schrödinger方程这一新的模型。本项目在研究不带限制与带调和限制的两类经典非线性Schrödinger方程基础上，发展新的变分方法研究新模型的整体动力学。我们以发展系统的不变集为导引，研究新模型整体解存在的门槛条件。探寻不同门槛之间的关系及整体解存在的最优不变集。确定在不同不变集内整体解的演化图景和渐进行为。研究具各种频率的驻波的存在性及在不同条件下驻波的稳定性。以Tao的孤立子分解猜想为导引，研究整体解的驻波分解。利用新模型的研究结果和三类方程之间的关系，进一步深化经典方程整体动力学的研究，深入揭示方程整体解与驻波相关联的动力学图景，推进非线性Schrödinger方程整体动力学的研究。关注新模型的物理实验和数值计算，实现对新的量子现象的数学描述和动态预测。

研究了带部分限制势的非线性Schrodinger方程以及相关带势非线性Schrodinger方程、双非线性幂非线性Schrodinger方程、Davey-Stewartson系统、随机Ginzburg-Landau方程以及随机波动系统的Cauchy问题。通过构造各种变分问题和交叉强制变分问题，提出了一套以现代变分分析理论为工具，研究非线性Schrodinger方程高速孤立子、小孤立子以及多孤立子存在性，进而讨论其对应驻波稳定性的新研究框架。结合Cauchy问题的适定性及各种对称不变性(守恒律), 深入研究了上述Cauchy问题解的整体动力学性质、爆破动力学性质以及孤立子动力学。在项目执行过程中，我们已发表研究论文18篇，其中16篇被SCI收录，包括：《J. Differential Equations》,《Advances in Nonlinear Analysis》,《J. Mathematical Physics》等。同时，培养毕业硕士研究生15名,博士生1名。