PhD research topic - Luca Fanelli
Spectral Theory & Dispersive PDE’s
Spectral stability, non self-adjoint Hamiltonians, nonlinear stability of solitons, uncertainty
principle, unique continuation
Some of the modern challenges of nonlinear PDE’s are related to the stability of some special solutions of soliton-type (solitons, kinks, breathers). In many models arising in fluid mechanics, a competition between dispersion and coservation produces soliton-like solutions, and their stability along the dynamics is an important fact. Among them, we propose to investigate about the Sasa-Satsuma model as an interesting mix between NLS and mKdV. In particular, unique continuations properties at two distinct times for solutions to this model does not seem to be known.
Another related theme is concerned wth the Zakharov-Kutsnetsov equation and the sharp space-decay at two distinct times of solutions.
In both cases, multipliers’ methods will come into play. We recently developed in [FKV] a rigid method which permits to get spectral stability for a large class of Schroedinger Hamiltonians, also in a non self-adjoint setting. We finally propose a thesis project about possible applications of the multipliers’ method to Hamiltonians definied on more general domains, such as waveguides and waveguides with bumps.
[SS] Sasa, Satsuma: New type of Soliton Solutions for a Higher-Order NLS (1991)
[FKV] Fanelli, Krejcirik, Vega: Spectral Stability for Schroedinger Hamiltonians with Subordinated Complex Potentials (2016)