Solvable Models of Nonlinear Dispersive Waves

描述流体、等离子体和光纤中的波的数学方程是极其复杂的。通常这些问题只能在高速计算机上解决。然而，通常可以用更简单的方程来近似这些复杂的问题，这些方程的解可以用显式解析形式表示。这样的方程被称为“可解模型”，并且多年来一直为人所知。一个例子是Korteweg-deVries方程，它模拟了水表面的波和等离子体中的声波。它是由Boussinesq于1877年发现的，但推导并不严谨。离子声等离子体方程的数值实验令人信服地表明，这种近似是稳健的——也就是说，该模型的有效性远远超出了它正式导出的小参数范围。这一奖项将支持对这一猜想进行数学上严格的证明。对其他这类模型的分析也将进行。其中包括由洛斯阿拉莫斯国家实验室的Camassa和Holm提出的浅水波浪模型方程，以及与Hele-Shaw流（两种不混溶液体在两个板块之间的流动）相关的可解模型。许多物理现象具有复杂的数学描述，在某些极限情况下可以简化为更简单的模型。一个例子是对海洋表面波的描述。完整的描述涉及复杂的方程组，必须在超级计算机上解决问题的每个特殊实例。然而，在浅水和单向波的重要特殊情况下，提出了一个更简单的近似方程。虽然这个简单的方程可以比任何数值模拟更详细、更彻底地研究，但问题是，对于哪些物理参数（在本例中为水深和波高）的近似描述确实是正确的。这些问题在过去曾引发过严重的科学争论，而且很少能通过严格的数学分析得到解决。他们的分辨率是非常理想的纯粹的智力原因和验证数值模拟。该奖项将支持在相关情况下（等离子体中的波）建立这种简化方程的有效性的工作，以及对描述流体流动现象的其他简化方程的研究。

Mathematical equations describing waves in fluids, plasmas, and optical fibers are extremely complex. Usually they can be solved only on high speed computers. However, it is often possible to approximate these complex problems with simpler equations whose solutions can be written in explicit analytical form. Such equations are called ``solvable models'', and have been known for years. An example is the Korteweg-deVries equation, which models waves of the surface of water and acoustic waves in plasmas. It was discovered in 1877 by Boussinesq, but the derivation was not rigorous. Numerical experiments on the ion acoustic plasma equations have shown convincingly that this approximation is robust - that is, that the validity of the model extends far beyond the small parameter range for which it was formally derived. This award will support a mathematically rigorous prooffor this conjecture. The analysis of other such models will also be carried out. Among these are model equations for shallow water waves thatwere proposed by Camassa and Holm of Los Alamos National Lab, and solvable models that are associated with Hele-Shaw flows (flows of two immiscible liquids between two plates).Many physical phenomena have complicated mathematical descriptions that canbe reduced to simpler models in certain limiting cases. An example is the description of surface waves in an ocean. The full description involves acomplicated system of equations that has to be solved on a supercomputerin each special instance of the problem. However, in the important special case of shallow water and unidirectional waves, a much simpler equation has been proposed as an approximation. While this simpler equation can be studied in great detail and much more thoroughly than with any numerical simulation, the question arises for which physical parameters (in this case waterdepth and wave height) the approximate description is indeed correct. Thesequestions have in the past led to serious scientific disputes, and there are very few cases where they have been resolved by rigorous mathematical analysis.Their resolution is very desirable for purely intellectual reasons and to validate numerical simulations. The award will support work to establish the validity of such reduced equations in a related situation (waves in plasmas)and the study of other reduced equations that describe fluid flow phenomena.