Bilinear control and growth of Sobolev norms for the nonlinear Schrödinger equation

We consider the nonlinear Schrödinger 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...

Full description

Saved in:
Bibliographic Details
Published inJournal of the European Mathematical Society : JEMS
Main Authors Duca, Alessandro, Nersesyan, Vahagn
Format Journal Article
LanguageEnglish
Published European Mathematical Society 10.01.2024
Subjects
Analysis of PDEs
Dynamical Systems
Mathematical Physics
Mathematics
Optimization and Control
Spectral Theory
37L55
approximate controllability
35R60
93B05 Nonlinear Schrödinger equation
geometric control theory
random perturbation
growth of Sobolev norms
81Q93
AMS subject classifications: 35Q55
Online AccessGet full text

Cover

More Information
Summary:We consider the nonlinear Schrödinger 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.
ISSN:1435-9855
1435-9863
DOI:10.4171/jems/1420