Asymptotic problems for some classes of dispersive PDEs

[excerpt from the introduction]: This work deals with some classes of nonlinear dispersive evolution PDEs: in particular, under some non classical framework we will consider a class of nonlinear Schrödinger equation, a class of nonlinear Klein-Gordon equation and a system of PDEs called Zakharov system which couples a Schrödinger-type equation with a nonlinear wave equation. For all of these equations the associated Cauchy problem will be considered. More specifically we will examine two different asymptotic limits: for the Schrödinger and the Klein-Gordon equations we will deal with the problem of scattering: roughly speaking, we wonder weather solutions to the nonlinear Cauchy problem behave as linear solutions for large times. The second asymptotic problem is instead a singular limit result related to the solution of the Zakharov system, which depends on a physical parameter: the qualitative investigation of the solutions for large value of such parameter is meaningful from a physical point of view. [...]