New asymptotics in non-linear quantum mechanics and shortwave diffraction

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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.