Nonlinear Wave Motion

9703850 Ablwitz研究涉及一类在非线性波动研究中产生的具有重要物理意义的非线性系统。这些研究包括一类新的局域多维解，它可以与拓扑数有关，即指数或缠绕数。与自对偶Yang-Mills方程相联系的是一类新的非线性常微分方程组，称为Darboux-Halphen(DH)型系统。这些方程可以通过将它们与某些线性单行问题联系在一起并进行分析来集成。线性问题具有正在演变的单行数据。DH系统的解可以是非常复杂的，在复杂的平面上表现出自然的边界和密集的分支。在特殊情况下，解用自同构函数表示。研究人员经常使用数值模拟来获得非线性波动方程解的近似值。PI和合作者的研究表明，截断和舍入误差可以导致计算混乱。可积系统的使用是非常重要的，因为分析结果可以与数值模拟相比较。非线性波动方程的研究在物理和工程中有着广泛的应用。与这一提议中的研究密切相关的是一类方程，称为可积系统。可积系统的数学及其解已被证明在我们理解各种现实世界现象中发挥着重要的和基本的作用。可积系统通常允许一类非常稳定的局域波解，称为孤子，这类解经常出现在物理环境中。例如，在流体动力学的研究中，在海洋的许多地方，例如俄勒冈州海岸附近，已经发现了巨大的孤立子内部海洋波，有时长度为一英里，振幅为数百英尺。这样的波对其路径上捕捉到的任何东西都可能是毁灭性的，因此了解这种内部孤子在海洋中的位置是有意义的。孤子的另一个应用是研究和设计高效的长距离通信系统。孤子消除了对电子信号再生器的需求，因为电子信号再生器会严重抑制数据速率。研究人员已经在实验室中成功地使用了孤子，目前它们正被用于原型通信系统。很有可能，未来的高数据速率通信系统将被设计成传播多个孤波。

9703850 Ablowitz Research involving a class of physically significant nonlinear systems arising in the study of nonlinear wave motion is proposed. Included in the studies are a new class of localized multidimensional solutions which can be related to a topological number; i.e. an index or winding number. Associated with the self dual Yang-Mills equations is a novel class of nonlinear ordinary differential equations, referred to as Darboux-Halphen (DH) type systems. These equations can be integrated by associating them with, and analyzing, certain linear monodromy problems. The linear problems have monodromy data which are evolving. The solutions of DH systems can be extremely complicated, exhibiting natural boundaries and dense branching in the complex plane. In special cases the solutions are expressed in terms of automorphic functions. Frequently researchers use numerical simulation to obtain approximations to solutions of nonlinear wave equations. Research by the PI and collaborators has demonstrated that computational chaos can be induced from truncation and roundoff errors. Use of integrable systems is extremely important since analytical results can be compared with numerical simulation. The study of nonlinear wave equations has wide ranging applications in physics and engineering. Intimately related to the studies in this proposal is a class of equations, called integrable systems. The mathematics of integrable systems and their solutions have been shown to play an important and fundamental role in our understanding of a variety of real world phenomena. Integrable systems usually admit a special class of extremely stable, localized wave solutions, referred to as solitons, which arise frequently in physical contexts. For example, in the study of fluid dynamics, huge soliton internal ocean waves, sometimes a mile in length, and hundreds of feet in amplitude, have been located in many parts of the ocean; e.g. near the coast of Oregon. Such waves can be devastating to anything caught in their path, hence knowledge of where such internal solitons are located in the ocean is of interest. Another application of solitons is in the study and designing of efficient long distance communication systems. Solitons eliminate the need for electronic signal regenerators which can seriously inhibit data rates. Researchers have used solitons successfully in the laboratory and currently they are being used in prototype communication systems. It may well be that future high data rate communications systems will be designed to propagate multiple soliton waves.