Uncertainty Principles for Nonlinear Wave Equations

In the case of a differential equation, an uncertainty principle applies when a solution to this equation cannot be very small (almost zero) at two different times, unless it vanishes completely for all time.
The aim of this project is to study uniqueness properties (uncertainty principles) of the solutions of different differential equations: the Korteweg—de Vries (KdV), modified KdV (mKdV), Camassa—Holm (CH), the Toda lattice (TL) and also the integrable non-linear Schrödinger (NLS) equation.
These equations play an important role in physics. The KdV and mKdV equations model the behavior of long waves on shallow water and the CH was introduced in 1993 to show that it was capable of modeling wave breaking, something that was unknown for the KdV equation. The Toda lattice is a simple model for a nonlinear one-dimensional crystal and it describes the motion of a chain of particles with nearest neighbor interaction. And finally the Schrödinger equation and also the NLS equation is a mathematical formulation for studying quantum mechanical systems. We will prove some uncertainty principles for these equations by relaxing the hypothesis on previous works. For example it is our intention to consider solutions that do not have to be compact support on a half line.
It is also our purpose to characterize the solutions of the NLS on graphs by knowing only part of it and we want to extend the uncertainty principle for the Schrödinger equation on graphs with web-like structure to potentials that can depend on time. These graphs consist of an inner part, formed by a finite number of vertices, and some finite number of semi-infinite chains attached to it. These systems appears for instance on the study of small oscillations of particles near its equilibrium position