My primary research interests
Many PDEs arising in physics can be seen as infinite dimensional Hamiltonian systems. The integrable ones, with the Korteweg-de Vries equation (KdV) and the 1-d nonlinear Schrödinger equation (NLS) as prominent examples, have a very rich structure and turn out to be important nonlinear models. One of the main tools in their analysis is the method of normal form: if these equations are expressed in suitable coordinates such as action-angle or Birkhoff coordinates, they can be solved by quadrature and the geometry of their phase space becomes much more tangible. I am particularly interested in the construction of these coordinates in low regularity regimes, and also in the analysis of the longterm behavior of solutions of perturbations of these equations using normal form theory.
Most results in integrable PDEs concern those who admit a Lax pair representation. Instead of analyzing the PDE directly, one can then focus on spectral theory of linear operators. Many qualitative features of solutions of integrable PDEs are encoded in the spectrum of its Lax operator. For example, in many cases regularity properties of solutions can be recovered from the asymptotic behavior of certain spectral quantities. In the last two decades mostly the self-adjoint cases were studied, such as for the real KdV and defocusing NLS. Recently, also non self-adjoint operators moved into consideration, such as for the complex KdV and the focusing NLS. I am particularly interested in the relation of instabilities for the focusing NLS and the singular behaviour of its periodic spectrum.