Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, New York, NY 10012;
Center for Atmosphere and Ocean Science, Courant Institute of Mathematical Sciences, New York University, New York, NY 10012.
Proc Natl Acad Sci U S A. 2019 Mar 5;116(10):3982-3987. doi: 10.1073/pnas.1820467116. Epub 2019 Feb 13.
Understanding and predicting extreme events and their anomalous statistics in complex nonlinear systems are a grand challenge in climate, material, and neuroscience as well as for engineering design. Recent laboratory experiments in weakly turbulent shallow water reveal a remarkable transition from Gaussian to anomalous behavior as surface waves cross an abrupt depth change (ADC). Downstream of the ADC, probability density functions of surface displacement exhibit strong positive skewness accompanied by an elevated level of extreme events. Here, we develop a statistical dynamical model to explain and quantitatively predict the above anomalous statistical behavior as experimental control parameters are varied. The first step is to use incoming and outgoing truncated Korteweg-de Vries (TKdV) equations matched in time at the ADC. The TKdV equation is a Hamiltonian system, which induces incoming and outgoing statistical Gibbs invariant measures. The statistical matching of the known nearly Gaussian incoming Gibbs state at the ADC completely determines the predicted anomalous outgoing Gibbs state, which can be calculated by a simple sampling algorithm verified by direct numerical simulations, and successfully captures key features of the experiment. There is even an analytic formula for the anomalous outgoing skewness. The strategy here should be useful for predicting extreme anomalous statistical behavior in other dispersive media.
理解和预测复杂非线性系统中的极端事件及其异常统计数据是气候、材料和神经科学以及工程设计中的一个重大挑战。最近在弱湍流水体中的实验室实验揭示了一个显著的转变，即表面波穿过突然的深度变化（ADC）时，从高斯行为转变为异常行为。在 ADC 的下游，表面位移的概率密度函数表现出强烈的正偏态，同时极端事件的水平升高。在这里，我们开发了一个统计动力学模型来解释和定量预测上述异常统计行为，同时实验控制参数发生变化。第一步是在 ADC 处使用时间匹配的输入和输出截断 Korteweg-de Vries（TKdV）方程。TKdV 方程是一个哈密顿系统，它诱导输入和输出统计 Gibbs 不变测度。在 ADC 处对已知的几乎高斯输入 Gibbs 态进行统计匹配完全确定了预测的异常输出 Gibbs 态，该态可以通过简单的采样算法计算，并通过直接数值模拟进行验证，成功捕捉到实验的关键特征。甚至还有一个关于异常输出偏度的解析公式。这里的策略应该对预测其他弥散介质中的极端异常统计行为有用。
