Decay of solutions of dispersive equations and Poisson brackets in algebraic geometry

Abstract

In the first part of this work we will study the spatial decay of solutions of nonlinear dispersive equations. The starting point will be the Korteweg-de Vries (KdV) equation, for which it will be proved that a decay of exponential type is degraded in time, and that the exhibited decay is optimal. In the second part we will make an exposition on Symplectic and Poisson Geometry with connections in Classical Mechanics to motivate a more abstract view of Poisson structures. With these preliminaries we can then give way to a little digression on Integrable Systems, and discuss the notion of complete integratbility in the sense of Liouville