On periodic solutions of damped-driven Maxwell–Bloch system

The invention of lasers in 1960 capped experimental work in the 1950s inspired by ideas from quantum physics. A wide variety of laser devices are constantly used in industry and everyday life: a red beam of light to control the closing of doors in every elevator and garage door, the finest laser beam operations and laser therapy, as well as drilling bizarrely shaped parts from the strongest metals - Laser devices do all this. In the 1960s, theoretical physicists proposed a number of mathematical models to describe the functioning of laser devices, and in subsequent years this theory was intensively developed in the works of leading physicists and mathematicians. However, these theoretical studies were not rigorous mathematically, which is due to the exceptional complexity of the corresponding Maxwell-Schrödinger equations and its various approximations (Maxwell-Bloch equations) that describe the functioning of lasers. As a result, the theoretical understanding of the functioning of these devices increasingly lagged behind the success of technical progress based on empirical concepts. As a result, laser design and manufacturing today is essentially empirical. The goal of this project is to develop the first rigorous mathematical theory for the Maxwell-Bloch equations. Specifically, we plan to prove i) the existence of time-periodic solutions to such equations, and moreover, ii) the existence of single-frequency solutions, which is one of main mysteries of the laser action. We have already established and published preliminary results in both of these directions. Intensive work is currently underway to improve the results and to develop other directions. This is just the beginning of the development of a future rigorous mathematical theory of lasers, which is intended to facilitate progress in their design and production. I also hope that the developed methods will be useful in the mathematical analysis of other high-frequency electron devices.