New Aspects of Soliton Theory

摘要，DMS 0505566，D Kaup，中央佛罗里达大学标题：孤子理论的新方面这个奖项支持的研究将解决非线性波和孤子理论尚未完全理解的方面。 特别是，它是发展新的分析范式解决非线性问题。 本研究将发展一种反散射变换（IST）来求解一般的Camassa-Holm方程及相关方程的初值问题。 目前的技术只允许一个处理有限的一类初值问题。 Camassa-Holm方程是一类流体力学问题的重要极限，它只能处理正位势。 然而，预测的演变涉及负电位也是相当重要的。 已知这种势可以模拟波浪破碎。 这项研究将扩大IST方法的力量，在任何地区的潜力可能是负的和/或积极的情况下。 这项研究开发的技术很重要，因为它们可以应用于其他重要和相关的，但更复杂的系统。 一个相关的和重要的非线性光学系统被称为“简并双光子传播”，这是一个非常快的非线性相互作用，通过它可以从激光束产生强烈的二次谐波辐射。 后一种系统在未来的非线性光学系统中具有潜在的应用，包括光学逻辑电路和开关。 该奖项还将支持非线性波的研究，以寻找方法和手段，为包含孤立波的不可积物理系统建模。 这个奖项是为了支持研究，以扩大我们的数学方法到物理制度仍然无法解决。它将集中在一个方程的求解方法，该方程描述了浅水表面在扰动时的行为。 选择这个方程的原因不一定只是为了增加我们对表面水波的理解，尽管这会发生。 更确切地说，之所以做出这样的选择，是因为这个方程的结构相对简单，而且是一类重要物理系统的核心。 开发解决这个核心系统的方法将允许人们立即将相同的技术应用于这些其他系统。 上面提到的这些系统之一来自光学物理学。 另一个这样的系统也存在于光学物理学中，但也出现在物理学的其他领域。 它是一种相互作用，使在介质或等离子体中传播的激光束可以变得不稳定，或者可以稳定。 作为这项研究的产物，我们将有方法来预测上述相互作用何时以及如何产生稳定或不稳定。 从这些拟议的研究课题的结果将有一个更广泛的影响，而不仅仅是提出的数学。它们将扩展可以用数学研究的物理系统，将为数学研究提供新的领域，并将进一步理解所研究的物理系统背后的物理学。

Abstract, DMS 0505566, D Kaup, University of Central FloridaTitle: New Aspects of Soliton TheoryThe research supported by this award will address aspects of nonlinear waves and soliton theory which have not be fully understood before. In particular, it is to develop new analytical paradigms for solving nonlinear problems. This research will develop an Inverse Scattering Transform (IST) for general initial value problems of the Camassa-Holm and related equations. Current techniques only allow one to treat a limited class of initial value problems. For the Camassa-Holm Equation, which is an important limit of a class of hydrodynamical problems, only positive potentials can be treated. However, the prediction of the evolution involving negative potentials are also quite important. Such potentials are known to model wave-breaking. This research will expand the power of the IST method to those cases where the potential could be negative and/or positive in any region. The techniques developed by this research are important because they then could be applied to other important and related, but more complex systems. One related and important nonlinear optical system is called "Degenerate Two-Photon Propagation", which is a very fast nonlinear interaction, by which one can generate intense second harmonic radiation from a laser beam. The latter system has potential applications to future nonlinear optical systems, including optical logic circuits and switches. This award will also support nonlinear wave research for finding methods and means for modeling nonintegrable physical systems which do contain solitary waves. This award is to support research to extend our mathematical methods into physical regimes which have remained unsolvable. It will center on a solution method for an equation which describes how the surface of shallow water behaves when disturbed. The reason for the choice of this equation is not necessarily just for increasing our understanding of surface water waves, although that will occur. Rather, the choice is made because this equation is relatively simple in its structure, and furthermore, is at the central core of a class of important physical systems. Developing methods for solving this core system will allow one to immediately apply the same techniques to these other systems. One of these systems, mentioned above, is from optical physics. Another such system is also found in optical physics, but also occurs in other areas of physics as well. It is an interaction whereby a laser beam, propagating in a medium or a plasma, can go unstable, or could stabilize. As a product of this research, we would then have methods for predicting when and how the above mentioned interaction would produce a stability or an instability. The results from these proposed research topics will have a broader impact than simply the mathematics that are proposed. They will expand the physical systems that can be studied mathematically, will provide new areas for mathematical research, and will further the understanding of the physics behind the physical systems so studied.