An overview on the standing waves of nonlinear Schrödinger and Dirac equations on metric graphs with localized nonlinearity. (English) Zbl 1416.35287
Summary: We present a brief overview of the existence/nonexistence of standing waves for the NonLinear Schrödinger and the NonLinear Dirac Equations (NLSE/NLDE) on metric graphs with localized nonlinearity. First, we 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 by the convergence of the bound states.
MSC:
| 35R02 | PDEs on graphs and networks (ramified or polygonal spaces) |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 81Q35 | Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices |
| 35Q40 | PDEs in connection with quantum mechanics |
| 49J40 | Variational inequalities |
| 49J35 | Existence of solutions for minimax problems |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
| 46T05 | Infinite-dimensional manifolds |
Keywords:
metric graphs; NLS; NLD; ground states; bound states; localized nonlinearity; nonrelativistic limitReferences:
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