Chapter 15 Hamiltonian PDEs

Abstract

This chapter discusses qualitative properties of solutions for Hamiltonian partial differential equations (PDE) in the finite volume case. Most of these properties have analogies in the classical finite-dimensional Hamiltonian mechanics. In the infinite-volume case, properties of the equations become rather different because of the phenomenon of radiation. The class of Hamiltonian PDEs contains many important equations of mathematical physics. The first difficulty one comes across when this class is absence of a general theorem that would guarantee that (locally in time) an equation has a unique solution is studied. Hamiltonian PDEs play the same role as its classical finite-dimensional counterpart plays for usual Hamiltonian equations; it is used to transform an equation to a normal form, usually in the vicinity of an invariant set.