Amplitude equations – an invitation to multi-scale analysis

Despite the steady increase in computing power many nonlinear partial differential equations still cannot be solved by brute-force numerical methods in acceptable time. However, often the method of multiple scales resulting in so-called amplitude equations can be used to first reduce the complexity of the problem. The amplitude equations give valuable mathematical insight and, moreover, can be treated numerically by orders of magnitude faster and hence be used to approximate physically or technologically interesting solutions with high accuracy. We explain this method using some simple examples, starting with ODEs and progressing to some pattern-forming systems and nonlinear wave equations.