Communications in Mathematical Sciences

Volume 19 (2021)

Number 2

The nonlinear Schrödinger equation with white noise dispersion on quantum graphs

Pages: 405 – 435

DOI: https://dx.doi.org/10.4310/CMS.2021.v19.n2.a5

Authors

Iulian Cîmpean (Faculty of Mathematics and Computer Science, University of Bucharest, Romania; and Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest, Romania)

Andreea Grecu (Faculty of Mathematics and Computer Science, University of Bucharest, Romania; and Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest, Romania)

Abstract

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.

Keywords

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

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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