# Nonlinear water waves and nonlocal model equations

## Abstract

The study of water waves is mathematically challenging, since they are described by a system of nonlinear partial differential equations in a domain with a free boundary. The aim of this project is to construct and study different types of solutions to these equations as well as to simplified nonlocal model equations. In particular, we will study travelling waves, that is, waves of permanent form which travel with constant speed. We will investigate singular travelling wave solutions of nonlocal model equations as well as three-dimensional solitary waves of the full water wave problem. We will also investigate stability properties of such waves, which is important for determining their physical relevance. In addition, we will investigate modulating pulses (or moving breathers). These consist of a permanent localised envelope which modulates an underlying periodic wave train which moves relative to the envelope. Such solutions have only been constructed for simpler model equations. Our aim is to prove the existence of modulating pulses in the full water wave problem. The analysis relies on bifurcation theory, inverse spectral theory, variational methods and infinite-dimensional dynamical systems theory. The questions that we plan to study will require the development of new mathematical tools in these areas.

## Team

- Erik Wahlén (PI)
- Tien Truong (PhD student)

## Funding source

Swedish Research Council (Vetenskapsrådet) grant No 2016-04999