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

该奖项将支持临界非线性薛定谔方程（NLE）的研究，即具有三次非线性的方程。我们感兴趣的是坍缩事件，即在有限时间内单点解的空间收缩。单个坍缩事件现在被很好地理解了，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。为此，必须对方程进行正则化，因为解不能在完全坍缩之后继续存在。要研究的问题之一是解决方案对正则化选择的依赖性。据推测，这种正则化只会对坍塌湍流产生适度的影响，这一猜想将在本项目中进行研究。在回溯到Kolmogorov的流体动力学方程的背景下，对于湍流的统计研究有一个一般的框架，而这项工作将把NLS解的坍缩湍流放在这个一般框架中。这个主题非常适合研究生训练，学生将得到其他大学和国家实验室的研究小组的支持和工作。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，在海洋学的异常波模型中。稳定的运动波（如异常波或光脉冲）被称为孤子，在NLS的解中，单个孤子的自发出现现在已经很好地理解了。这项工作将研究这种孤子随机出现和不可预测的情况，但仍然遵循统计模式。该奖项所做的工作将有助于理解这些统计模式。这种现象类似于紊流，紊流也具有统计模式下的不可预测性。该奖项还将支持在这个令人兴奋和广阔的领域培养学生。

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.