New asymptotics in non-linear quantum mechanics and shortwave diffraction

非线性量子力学和短波衍射中的新渐近论

• 批准号: 261412-2006
• 负责人: Kondratieva, Margrita
• 金额: $ 0.72万
• 依托单位: Memorial University of Newfoundland
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=363294
• 关键词: New; asymptotics; non; linear; quantum

Most people are familiar with the appearance of waves on the sea. The fascination of watching  waves at some distance from the shore advance towards you never seems to fade. A scientist watching the waves will wonder about their origin and behaviour. In fact, the motion of waves  can be modeled by  mathematical equations and studied by  appropriate mathematical methods. Moreover, it has been found that our life is surrounded by waves of different natures. Acoustic and electro-magnetic waves provide our natural ability to hear and see each other. Radios,  TVs, microwave ovens, optical cables, cell phones and communication equipment are built because of our understanding of wave propagation. In the development of the quantum theory of the micro-universe a generalisation of the concept of a moving wave played a key role. This theory uses so-called probabilistic waves which describe the likelihood of finding a micro-particle moving in a certain domain with a certain speed at a given moment of time. In this research program we  study two fundamental problems. The first is a mathematical equation, which has been extensively applied in modern nonlinear optics, quantum electrodynamics, modeling the  laser beam propagation,  and Bose-Einstein condensation which emerged in  an array of recent physical experiments. We will develop an approximate method of solving this equation which will allow us to understand these phenomena in greater detail. The second problem is about  electromagnetic  wave diffraction from  an obstacle. The method we will develop will allow us to write a computer code for a numerical simulation, for instance,  of micro-optical devices. This is important in optimising the design and construction process. In the case when an exact solution is not available, asymptotical methods allow one to find an approximate solution of an equation in some range of physical parameters (such as mass, size, frequency, etc.). Due to the complexity of the equations which we consider, development of new asymptotic methods is often an irreplaceable tool of theoretical investigation.

大多数人都熟悉海上波浪的外观。从海岸前进到一段距离的海浪的迷恋似乎永远不会消失。观看海浪的科学家会想知道它们的起源和行为。实际上，波的运动可以通过数学方程式建模，并通过适当的数学方法进行研究。此外，已经发现我们的生活被不同的本性浪潮所包围。声学和电磁波提供了我们自然的声音和彼此见面的能力。由于我们对波浪传播的理解，建造了无线电，电视，微波炉，光电缆，手机和通信设备。在微宇宙的量子理论的发展中，移动波的概念的概括起着关键作用。该理论使用所谓的概率波，描述了在给定时间时刻以一定速度移动的微粒子在某个域中移动的微粒子的可能性。在该研究计划中，我们研究了两个基本问题。第一个是数学方程，它已广泛应用于现代的非线性光学元件，量子电动力学，对激光束传播建模和Bose-Einstein凝结，这些凝结物在最近的一系列物理实验中出现。我们将开发一种求解该方程式的近似方法，这将使我们能够更详细地了解这些现象。第二个问题是关于障碍物的电磁波衍射。我们将开发的方法将使我们能够为数值模拟，例如微光学设备编写计算机代码。这对于优化设计和施工过程很重要。在没有精确解决方案的情况下，不对称方法允许在某些物理参数（例如质量，大小，频率等）中找到方程的近似解。由于我们考虑的方程式的复杂性，新的不对称方法的发展通常是理论投资的不可替代的工具。