Geometric Numerical Integration and Schrödinger Equations

摘要

Geometric numerical integration is concerned with the question of whether numerical methods are able to correctly reproduce qualitative properties of differential equations over long times. Over the past 20 years, this has been a very active topic of research in numerical analysis. The present book is centered around the question to what extend we can reliably simulate the long time behavior of Hamiltonian partial differential equations with numerical methods. For the sake of a simple presentation, it focuses on the linear and certain nonlinear Schrödinger equations, subject to periodic boundary conditions. References to more general situations, however, are given. The book starts from some enlightening examples which display possible instability phenomena, mainly due to resonances. The remaining chapters then serve to rigorously explain these numerical observations with the help of a sound mathematical theory. In particular, mutual energy exchanges are studied that take place between the modes in cubic nonlinear Schrödinger equations.