New asymptotics in non-linear quantum mechanics and shortwave diffraction

大多数人都熟悉大海上波浪的样子。看着海岸远处的海浪向你袭来的魅力似乎永远不会消退。观察海浪的科学家会想知道它们的起源和行为。事实上，波的运动可以用数学方程来模拟，并可以用适当的数学方法来研究。此外，人们发现我们的生活被不同性质的波浪包围着。声波和电磁波为我们提供了互相听到和看到对方的自然能力。收音机、电视、微波炉、光缆、手机和通信设备都是由于我们对波传播的理解而制造出来的。在微观宇宙量子理论的发展中，运动波概念的推广起了关键作用。这个理论使用了所谓的概率波，它描述了在特定时刻以特定速度在特定区域内移动的微粒的可能性。在这个研究项目中，我们研究两个基本问题。第一个是数学方程，它被广泛应用于现代非线性光学、量子电动力学、激光传播建模以及近年来一系列物理实验中出现的玻色-爱因斯坦凝聚。我们将发展一种近似解这个方程的方法，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允许我们为数值模拟编写计算机代码，例如，微光学设备。这对于优化设计和施工过程非常重要。在没有精确解的情况下，渐近方法允许人们在一定范围的物理参数（如质量、尺寸、频率等）中找到方程的近似解。由于我们所考虑的方程的复杂性，发展新的渐近方法往往是理论研究的不可替代的工具。

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.