Collaborative Research: Strong Turbulence from Singular Collapses in Nonlinear Schroedinger Type of Equations

该奖项将支持对临界非线性薛定谔方程（NLE）的研究，即具有立方非线性的方程。感兴趣的是崩溃事件，即在有限时间内单点的解决方案的空间收缩。现在，人们已经很好地理解了单个坍缩事件，这项工作将研究坍缩湍流，即表现出坍缩事件随机分布的解。为了这个目的，方程必须正则化，因为解不能在完全坍缩之后继续。然后要研究的问题之一是解决方案对正则化选择的依赖性。我们认为这种正则化对坍缩湍流的影响是有限的，本项目将对此进行研究。有一个一般的框架，湍流的统计研究的背景下，流体动力学方程，可以追溯到Kolmogorov，这项工作将崩溃湍流的解决方案的NLS在这个一般的框架。该课题非常适合研究生培养，学生将获得其他大学和国家实验室研究小组的支持和接触。非线性薛定谔方程（NLS）描述了波随时间的非线性相互作用，是非线性科学中的通用模型。它出现在激光聚变的描述中，在光纤中，在海洋学中的流氓波模型中。稳定的运动波（如流氓波或光脉冲）被称为孤子，现在已经很好地理解了NLS解中单个孤子的自发出现。这项工作将研究这种孤子随机和不可预测地出现，但仍然遵循统计模式的情况。该奖项所做的工作将有助于理解这些统计模式。这种现象类似于湍流流体流动，其特征也在于以统计模式发生的不可预测性。该奖项还将支持在这个令人兴奋且广泛的领域对学生进行培训。

This award will support research on critical nonlinear Schroedinger equations (NLE), i.e. equations with a cubic nonlinearity. Of interest are collapse events, that is, spatial contractions of solutions to single points in finite time. Individual collapse events are now well understood, and this work will study collapse turbulence, that is, solutions that exhibit random distributions of collapse events. For this purpose, the equation has to be regularized, since solutions cannot be continued beyond a complete collapse. One of the issues to be studied then is the dependence of solutions on the choice of regularization. It is conjectured that this regularization will only have a moderate effect on collapse turbulence, and this conjecture will be studied in this project. There is a general framework for the statistical study of turbulence in the context of the equations of fluid dynamics that goes back to Kolmogorov, and this work will place collapse turbulence of solutions of the NLS in this general framework. The topic is very suitable for graduate training, and students will be supported and exposed to work done by research groups at other universities and at national laboratories.The nonlinear Schrodinger equation (NLS), which describes the nonlinear interaction of waves over time, is a universal model in nonlinear science. It occurs in the description of laser fusion, in fiber optics, and in models for rogue waves in oceanography. Stable moving waves (such as rogue waves or light pulses) are called solitons, and the spontaneous emergence of individual solitons in solutions of the NLS is now well understood. This work will study situations where such solitons appear randomly and unpredictably, but still following statistical patterns. The work done with this award will contribute to the understanding of these statistical patterns. This phenomenon is similar to turbulent fluid flow, which is also characterized by unpredictability that occurs with a statistical pattern. The award will also support the training of students in this exciting and broad field.