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。令人感兴趣的是崩塌事件，即在有限时间内单点解的空间收缩。现在人们已经很好地了解了单个坍塌事件，这项工作将研究坍塌湍流，即呈现坍塌事件随机分布的解。为此，方程必须正则化，因为在完全崩溃之后，解不能继续。然后要研究的问题之一是解对正则化选择的依赖性。据推测，这种正则化只会对崩塌湍流有适度的影响，这一猜想将在本项目中进行研究。在可追溯到柯尔莫戈洛夫的流体动力学方程的背景下，有一个关于湍流的统计研究的一般框架，这项工作将把NLS解的崩溃湍流放在这个一般框架中。这一课题非常适合研究生培养，学生将得到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，是非线性科学中的一个普遍模型。它出现在激光聚变的描述中，出现在光纤中，出现在海洋学中的无赖波模型中。稳定运动的波(如无赖波或光脉冲)被称为孤子，而在NLS的解中自发出现的单个孤子现在已经被很好地理解了。这项工作将研究这样的孤子随机和不可预测地出现，但仍然遵循统计模式的情况。利用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，后者的特征也是统计模式下发生的不可预测性。该奖项还将支持学生在这一令人兴奋和广泛的领域的培训。

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.