Bilinear control and growth of Sobolev norms for the nonlinear Schr\"odinger equation

TL;DR

This paper demonstrates approximate controllability of the nonlinear Schrödinger equation on a torus using a time-dependent external potential, and explores the long-term unbounded growth of solutions with random potentials.

Contribution

It introduces a multiplicative geometric control method for NLS and applies it to analyze the behavior of solutions under random potentials.

Findings

NLS is approximately controllable between eigenstates in small time.

Solutions with random potentials are almost surely unbounded in Sobolev spaces.

Develops a new control approach for nonlinear PDEs.

Abstract

We consider the nonlinear Schr\"odinger equation (NLS) on a torus of arbitrary dimension. The equation is studied in presence of an external potential field whose time-dependent amplitude is taken as control. Assuming that the potential satisfies a saturation property, we show that the NLS equation is approximately controllable between any pair of eigenstates in arbitrarily small time. The proof is obtained by developing a multiplicative version of a geometric control approach introduced by Agrachev and Sarychev. We give an application of this result to the study of the large time behavior of the NLS equation with random potential. More precisely, we assume that the amplitude of the potential is a random process whose law is $1$-periodic in time and non-degenerate. Combining the controllability with a stopping time argument and the Markov property, we show that the trajectories of the random equation are almost surely unbounded in regular Sobolev spaces.