"The motivation for a physicist to study 1-dimensional problems is best illustrated by the story of the man who, returning home late at night after an alcoholic evening, was scanning the ground for his key under a lamppost; he knew, to be sure, that he had dropped it somewhere else, but only under the lamppost was there enough light to conduct a proper search."
-Calogero, 1971 (humorously)
I work in partial differential equations and harmonic analysis, specifically completely integrable systems and soliton dynamics.
Completely integrable systems can be viewed as a natural generalization of solvable (integrable) classical systems. Since physicists first used math to study physical systems, there has been a search for explicit formulas to describe motion. However, as systems and settings become more complicated, the notion of explicit formula needs to generalize as well. We then say that an n-dimensional system is integrable if there exists n "independent" conserved quantities. These n conserved quantities allow us to define an explicit solution, albeit in a potentially unnatural form. Perhaps the most famous integrable system is the Calogero-Moser particle system:
a system of n particles on a line interacting via an inverse cubed force. This system was introduced by Calogero, solved in the quantum sense by Calogero and Marchioro, and then solved in the classical sense by Moser. When discussing the motivation behind studying such an equation, Calogero humorously referenced the story of the drunkard as quoted above.
Since partial differential equations are infinite-dimensional, we define a completely integrable PDE as one that has an infinite hierarchy of "independent" conserved quantities. These conserved quantities, as well as additional structure not discussed here, makes completely integrable PDEs open to deep exploration and results. At the moment, I am studying the focusing Continuum Calogero-Moser Model, otherwise known as the Calogero-Moser derivative NLS:
I am working with complex-valued solutions u(t,x) on the real line which belong to a Hardy-Sobolev space.
