NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2017, 8 (1), P. 29–37
Nonlinear standing waves on planar branched systems: shrinking into metric graph
Z. Sobirov – Tashkent Financial Institute, 60A, Amir Temur Str., 100000, Tashkent, Uzbekistan; firstname.lastname@example.org
D. Babajanov – Turin Polytechnic University in Tashkent, 17 Niyazov Str., 100095, Tashkent, Uzbekistan
D. Matrasulov – Turin Polytechnic University in Tashkent, 17 Niyazov Str., 100095, Tashkent, Uzbekistan
We treat the stationary nonlinear Schrödinger equation on two-dimensional branched domains, so-called fat graphs. The shrinking limit when the domain becomes one-dimensional metric graph is studied by using analytical estimate of the convergence of fat graph boundary conditions into those for metric graph. Detailed analysis of such convergence on the basis of numerical solution of stationary non-linear Schrödinger equation on a fat graph is provided. The possibility for reproducing different metric graph boundary conditions studied in earlier works is shown. Practical applications of the proposed model for such problems as Bose-Einstein condensation in networks, branched optical media, DNA, conducting polymers and wave dynamics in branched capillary networks are discussed.
Keywords: metric graph, Schrödinger equation.
PACS 05.45.Yv, 42.65.Wi, 42.65.Tg