An overview on the standing waves of nonlinear Schrödinger and Dirac equations on metric graphs with localized nonlinearity
arXiv:1901.02696 · doi:10.3390/sym11020169
Abstract
We present a brief overview on the existence/nonexistence of standing waves for the NonLinear Schrödinger 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.
20 pages, 10 figures; keywords: metric graphs, NLS, NLD, ground states, bound states, localized nonlinearity, nonrelativistic limit