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

具有非线性力的波动方程存在于许多应用领域，如光学或量子力学。在这个项目中，我们研究了当系统受到低水平的阻尼时，这类方程的解的行为。第二个方面是存在强大的陀螺力，例如科里奥利力、外部磁场或相对论量子力学的非相对论极限。要分析这些极限，我们需要适应问题的数学技术；要模拟方程，我们需要具有渐近性的数值格式。我们首先从数学分析的角度研究阻尼力和陀螺力，然后以开发新的更有效的数值格式为目标。作为一个主要因素，我们开发了一种新的方法，用于计算包含多个高频项的积分，预计这将超出我们特定的应用领域。

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.