On semilinear wave equations - damping, asymptotics, and exponential integrators in the highly oscillatory regime

点击翻译按钮获取中文摘要

Wave equations with nonlinear forces occur in many fields of application, such as optics or quantum mechanics. In this project, we study the behavior of solutions to such equations when the system is subject to a low level of damping. A second aspect is the presence of strong gyroscopic forces, e.g. Coriolis forces, external magnetic fields, or the non-relativistic limit of relativistic quantum mechanics. To analyze these limits, we need problem adapted mathematical techniques; to simulate the equations, we need asymptotics-aware numerical schemes. We study damping and gyroscopic forces first from the the point of view of mathematical analysis, then with the goal of developing new and more efficient numerical schemes. As a main ingredient, we develop a new method for the evaluation of integrals with terms involving multiple high frequencies which is expected to be relevant beyond our specific domain of application.