Solvable Models of Nonlinear Dispersive Waves

描述流体、等离子体和光纤中波的数学方程极其复杂。通常只有在高速计算机上才能解决这些问题。然而，通常可以用更简单的方程来近似这些复杂的问题，这些方程的解可以写成显式的解析形式。这样的方程被称为“可解模型”，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在1877年发现的，但推导并不严格。对离子声等离子体方程的数值实验已经令人信服地表明，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推导的小参数范围。这一奖项将为这一猜想提供严格的数学证明。还将对其他此类模型进行分析。其中包括洛斯阿拉莫斯国家实验室的Camassa和Holm提出的浅水波模型方程，以及与Hele-Shaw流(两个不相容的液体在两个平板之间的流动)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。一个例子是对海洋中表面波的描述。完整的描述涉及一个复杂的方程组，在问题的每个特殊情况下，都必须在超级计算机上求解。然而，在浅水和单向波的重要特殊情况下，提出了一个简单得多的方程作为近似。虽然这个更简单的方程可以比任何数值模拟更详细和更彻底地研究，但问题是，对于哪些物理参数(在这种情况下，水深和波高)，近似描述确实是正确的。这些方程在过去曾导致严重的科学争议，很少有案例通过严格的数学分析来解决。出于纯粹的智力原因和验证数值模拟的原因，它们的解决是非常可取的。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.