## Integrable Systems

An Honours Course in Applied Mathematics

The mathematical theory of integrable systems has been described as one of the most profound advances of twentieth century mathematics. They lie at the boundary of mathematics and physics and were discovered through a famous paradox that arises in a model devised to describe the thermal properties of metals (called the Fermi-Pasta-Ulam paradox).

In attempting to resolve this paradox, Kruskal and Zabusky discovered exceptional properties in the solutions of a non-linear PDE, called the Korteweg-de Vries equation (KdV). These properties showed that although the solutions are waves, they interact with each other as though they were particles, i.e., without losing their shape or speed, until then thought to be impossible for solutions of non-linear PDEs. Kruskal invented the name

*solitons*for these solutions. Here is a picture of two solitons interacting.

Solitons are known to arise in other non-linear PDEs and also in partial difference equations. These systems and their symmetry reductions are now called *integrable systems*. These systems occur as universal limiting models in many physical situations.

This course introduces the inverse scattering transform method for solving such systems.

- References
- P.G. Drazin and R.S. Johnson
*Solitons : an introduction*Cambridge University Press, Cambridge, UK, 1989. - M. J. Ablowitz and H. Segur,
*Solitons and the inverse scattering transform*SIAM, Philadelphia, USA, 1981. - M. J. Ablowitz and P.A. Clarkson,
*Solitons, nonlinear evolution equations and inverse scattering*Cambridge University Press, Cambridge, UK, 1991.

- P.G. Drazin and R.S. Johnson
- Interesting Links on Solitons:
- An account of John Scott Russell’s discovery of “that singular and beautiful phenomenon, which I have called the wave of translation.”
- A modern attempt by mathematicians to recreate Scott Russell’s wave in the Union Canal near Edinburgh.
- The wikipedia page on Solitons (whose first and second definitions I disagree with).
- The wikipedia page on the Korteweg-de Vries equation.