Abstract

The lectures will provide an introduction to the theory of “modulation” and its role in the derivation of model equations, such as the KdV equation, Boussinesq equation, KP equation, and Whitham modulation equations, and their role in the theory of water waves. The classical theory of modulation, such as Whitham modulation theory, will be reviewed, and a new approach will be introduced. The new approach is based on modulation of background flow. Methodology that is key to the theory is symmetry and conservation laws, relative equilibria, Hamiltonian and Lagrangian structures, multiple scale perturbation theory, and elementary differential geometry. By basing the theory on modulation of relative equilibria, new settings are discovered for the emergence of KdV and other modulation equations. For example, it is shown that the KdV equation can be a valid model for deep water as well as shallow water. The lectures are introductory, and no prior knowledge is assumed.