Communications in Mathematical Sciences
Volume 19 (2021)
The nonlinear Schrödinger equation with white noise dispersion on quantum graphs
Pages: 405 – 435
We show that the nonlinear Schrödinger equation (NLSE) with white noise dispersion on quantum graphs is globally well-posed in $L^2$ once the free deterministic Schrödinger group satisfies a natural $L^1 - L^\infty$ decay, which is verified in many examples. Also, we investigate the well-posedness in the energy domain in general and in concrete situations, as well as the fact that the solution with white noise dispersion is the scaling limit of the solution to the NLSE with random dispersion.
quantum graphs, Schrödinger operator, white noise dispersion, Strichartz estimates, nonlinear Schrödinger equation, stochastic partial differential equations, spectral theory, nonlinear fiber optics
2010 Mathematics Subject Classification
34B45, 35J10, 35P05, 35Q55, 60H15, 81Q35, 81U30
The second-named author was partially supported by PhD fellowships of the University of Bucharest and L’Agence Universitaire de la Francophonie, and by a BITDEFENDER Junior Research Fellowship from the “Simion Stoilow” Institute of Mathematics of the Romanian Academy.
Received 8 November 2019
Accepted 11 September 2020
Published 12 April 2021