This study has led to the investigation of model problems for water waves, and in particular to the discovery of a class of Hamiltonian systems, common in the theory of waves, where solitary and periodic waves and bores occur as stable phenomena. These systems are studied using a tool not previously associated with Hamiltonian systems, namely the topological degree. These equations lie at the heart of many problems in physics, engineering, and mathematical biology. We are particularly interested in problems where the solutions of the equations develop singularities in a finite time. This behaviour is typical of models of combustion and burning which provide examples of problems with finite time blow-up, extinction or quenching, all of which are abstractions of observed physical phenomena.
Singularities also arise in certain theories of the nature of turbulence.
Describing these singularities requires the development of novel analytic and numerical tools and leads to many new mathematical problems. All rights reserved.