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 Kortewegde Vries (KdV), modified KdV (mKdV), CamassaHolm (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.

We have obtained various uncertainty principles for different partial differential equations (PDEs). As particular cases we have studied some PDEs which play an important role in physics: the Korteweg-de Vries equation and the modified Korteweg-de Vries equation, which model the behavior of long waves on shallow water, and the non-linear Schrödinger (NLS) equation, which is a mathematical formulation for studying quantum mechanical systems. Thus, what we have proved is that if we have a solution to one of these equations such that it is too small at two different times, then the solution is trivial, that is, it vanishes for all time. This is what we called an uncertainty principle for PDEs. We have also managed to extend this kind of result for more general PDEs. These equations must fullfill certain shape which allows one to apply the scattering theory. Actually this is one of the main points to prove the result. In particular, we combine the scattering theory and complex analysis. Moreover the measure of "how small" the solution should be has been studied by other authors, but they have used other techniques based on real analysis. This opens a new point of view to try to obtain more different uncertainty principles for other equations.