We discuss the wellposedness theory of the Cauchy problem for the nonlinear Schrödinger equation on compact Riemannian manifolds. New dispersive estimates on the
linear Schrödinger group are used to get global existence in the energy space on arbirary surfaces and three-dimensional manifolds, generalizing...
We review a new method for studying boundary value problems for evolution PDEs.
This method yields explicit results for a large class of evolution equations which iclude: (a) Linear equations with constant coefficients, (b) certain classes of linear
equations with variable coefficients, and (c) integrable...
The derivative nonlinear Schrödinger equation is shown to be locally well-posed in
a class of functions analytic on a strip around the real axis. The main feature of the
result is that the width of the strip does not shrink in time. To overcome the derivative
loss, Kato-type smoothing results and...
In this contribution, we describe the simplest, classical problem in water waves, and
use this as a vehicle to outline the techniques that we adopt to analyse this particular
approach to the derivation of soliton-type equations. The surprise, perhaps, is that
such an apparently transparent set of...
A general structure is developed from which a system of integrable partial difference
equations is derived generalising the lattice KdV equation. The construction is based
on an infinite matrix scheme with as key ingredient a (formal) elliptic Cauchy kernel.
The consistency and integrability of the...