Visan Group

Integrable and Non-integrable Hamiltonian PDEs

A prism splits light because different colors travel at different speeds. This phenomenon, known as dispersion, is ubiquitous. It appears naturally in optics, magnetohydrodynamics, quantum mechanics, and in the context of both surface and internal waves in fluid mechanics. Such wave propagation is modeled by dispersive partial differential equations (dispersive PDEs), which are the focus of Vișan’s group. Much of our intuition and understanding of such models has been driven by a specific subclass known as completely integrable systems, whose rich algebraic structures have allowed researchers to see more deeply. For example, while first discovered in the setting of completely integrable systems, solitons and multi-solitons have since found numerous applications in the applied sciences: in fiber optics, solitons have been employed in the transmission of digital signals over long distances, while in biology, they are used to describe signal propagation in the nervous system and low-frequency collective motion in proteins.

One of the most basic questions we can ask of a physical model is whether it makes testable predictions. Mathematicians have distilled this concept into the notion of well-posedness. A major focus of Vișan’s group is understanding the well-posedness of central models in dispersive PDEs. We also work to understand the stability or instability of special structures, such as solitons and multi-solitons, and to elucidate the long-time behavior of general solutions. Another facet of the group’s work is the construction of dynamics for wave systems in thermal equilibrium. Central to the group’s success has been a new synthesis of harmonic analysis and Hamiltonian mechanics.