Solvable Models of Nonlinear Dispersive Waves

非线性色散波的可解模型

• 批准号: 9971249
• 负责人: David Sattinger
• 金额: $ 0.87万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-01 至 1999-08-19
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971249&HistoricalAwards=false
• 关键词: Solvable; Models; Nonlinear; Dispersive; Waves

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.

描述流体，等离子和光纤中波的数学方程式非常复杂。 通常，它们只能在高速计算机上解决。但是，通常可以将这些复杂问题与更简单的方程式近似，其解决方案可以以显式的分析形式编写。这样的方程称为``可解决的模型''，多年来一直知道。一个例子是Korteweg-devries方程，该方程模拟了等离子体中水和声波表面的波。 它是Boussinesq于1877年发现的，但派生并不严格。 离子声学等离子体方程的数值实验令人信服地表明，该近似值是可靠的 - 也就是说，模型的有效性远远超出了其正式得出的小参数范围。该奖项将支持数学上严格的猜想。对其他此类模型的分析也将进行。其中包括由Los Alamos National Lab的Camassa和Holm提出的浅水波的模型方程，以及与Hele-Shaw流有关的可解决模型（两个板之间的两个不混溶的液体的流动）。许多物理现象之间具有复杂的数学描述，使其在某些限制案例中可以简化为简单模型。 一个例子是对海洋表面波的描述。 完整的描述涉及必须在问题的每个特殊实例上求解的方程式系统。但是，在浅水和单向波的重要特殊情况下，提出了一个更简单的方程式作为近似值。 尽管可以详细研究这个更简单的方程式，并且比任何数值模拟都更详细地研究了，但出现了一个问题，即在哪些物理参数（在这种情况下为WaterDepth和Wave Height）的问题确实正确。 这些问题过去引起了严重的科学纠纷，在很少有情况下，通过严格的数学分析解决了它们。由于纯粹的智力原因，他们的解决方案是非常需要的，并且可以验证数值模拟。 该奖项将支持在相关情况下（等离子体中的波浪）以及对描述流体流现象的其他简化方程的研究，以确定此类降低方程的有效性。