I first got in contact with Hamiltonian PDEs when I wrote my first thesis about the Nash-Moser-Theorem as some books mentioned them as a field of application of this theorem. As those examples were quite interesting, I would like to learn more about that area and the problems relating to them. However, it is hard to find introductory books or papers on Hamiltonian PDEs and when I do, I feel like hitting a wall after some time as they are too advanced.
Does anybody know a good introduction to Hamiltonian PDEs or what I could study beforehand to obtain the necessary background? I've already taken an undergraduate course on dynamical systems and some graduate courses on (nonlinear) functional analysis and a first course on PDEs (95% elliptic theory). Next term I will take a course on evolution equations (mostly semigroup-theory I guess) and theoretical mechanics. As some authors seem to use some methods from differential geometry, is it also useful to study that too, or is differential geometry not that necessary?