*Title*: **On Analysis of Integrable Evolution Equations**

*Abstract*: Given a partial differential equation (PDE), supplemented with appropriate data, three basic questions arise: (a) Existence of solutions in a good space;
(b) Uniqueness of solution in this space; (c) Stability of the solution, which means that small errors in the data result in small errors in the solution.
The study of these questions is central in the theory of PDE and is of great importance for equations modeling physical situations.
In this talk we shall analyze these questions for several evolution equations that arise in mathematical physics and are integrable,
like the nonlinear Schrödinger equation, the Korteweg de-Vries equation, and the Camassa-Holm equation.
The talk is based on work in collaboration with A. Fokas, C. Kenig, C. Holliman, D. Mantzavinos, G. Misiolek and G. Petronilho.

Smith Colloquium Spring 2018