We shall begin this talk by discussing the Fokas Unified Transform Method for solving the initial-boundary value problem (ibvp) of nonlinear evolution equations, like the Korteweg-de Vries equation and the nonlinear Schrodinger equation. Although introduced as the ibvp analogue of the renowned Inverse Scattering Transform method for integrable nonlinear evolution equations, Fokas’ method can also be used to produce novel solution formulas for the linear versions of such equations. Replacing in Fokas’ solution formulas the forcing with the nonlinearity provides a new framework for the analysis of nonlinear equations with a variety of boundary conditions in appropriate solution spaces, like Bourgain spaces.

Then, we shall discuss questions of existence, uniqueness, dependence on initial data, and regularity of solutions to the initial value problem of Camassa-Holm and related equations in a variety of function spaces. Some of these equations can be thought as “toy” models for the Euler equations governing the motion of an incompressible fluid, and the analytic techniques developed for these equations have been in some cases transferable to the Euler equations.