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

合作研究：非线性薛定谔方程中奇异塌陷引起的强湍流

• 批准号: 0807131
• 负责人: Pavel Lushnikov
• 金额: $ 10.84万
• 依托单位: University of New Mexico
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-01 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0807131&HistoricalAwards=false
• 关键词: Collaborative; Research; Strong; Turbulence; Singular

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.

该奖项将支持有关关键非线性施罗丁格方程（NLE）的研究，即具有立方非线性的方程式。感兴趣的是崩溃事件，即在有限时间内对单点的空间收缩。现在已经对个体崩溃事件进行了充分的了解，这项工作将研究崩溃的湍流，即表现出崩溃事件随机分布的解决方案。为此，由于无法在完全崩溃之外继续进行解决方案，因此方程式必须正规化。那么要研究的问题之一是解决方案对正则化的选择的依赖性。据推测，这种正则化只会对崩溃的湍流产生适度的影响，并且该猜想将在该项目中进行研究。在流体动力学方程的背景下，有一个统计研究的一般框架，该方程可以追溯到Kolmogorov，这项工作将在此一般框架中将NLS解决方案的崩溃湍流放置。该主题非常适合研究生培训，学生将受到支持和暴露于其他大学和国家实验室的研究小组所做的工作。非线性Schrodinger方程（NLS）描述了随着时间的推移，波浪的非线性相互作用是非线性科学的通用模型。它发生在激光融合，光纤和海洋研究中流氓波的模型中。稳定的移动波（例如流氓波或光脉冲）称为孤子，现在已经充分了解了NLS溶液中单个孤子的自发出现。这项工作将研究这些孤子随机和不可预测的情况，但仍遵循统计模式。通过此奖项完成的工作将有助于理解这些统计模式。该现象类似于湍流流，其特征在于具有统计模式的不可预测性。该奖项还将支持在这个令人兴奋和广泛的领域中对学生的培训。