An overview on the standing waves of nonlinear Schr\"odinger and Dirac equations on metric graphs with localized nonlinearity
Analysis of PDEs
2019-02-06 v1 Mathematical Physics
Functional Analysis
math.MP
Abstract
We present a brief overview on the existence/nonexistence of standing waves for the NonLinear Schr\"odinger and the NonLinear Dirac Equations (NLSE/NLDE) on metric graphs with localized nonlinearity. We first focus on the NLSE, both in the subcritical and the critical case, and then on the NLDE, highlighting similarities and differences with the NLSE. Finally, we show how the two equations are related in the nonrelativistic limit, proving the convergence of bound states.
Keywords
Cite
@article{arxiv.1901.02696,
title = {An overview on the standing waves of nonlinear Schr\"odinger and Dirac equations on metric graphs with localized nonlinearity},
author = {William Borrelli and Raffaele Carlone and Lorenzo Tentarelli},
journal= {arXiv preprint arXiv:1901.02696},
year = {2019}
}
Comments
20 pages, 10 figures; keywords: metric graphs, NLS, NLD, ground states, bound states, localized nonlinearity, nonrelativistic limit