Nonlinear dynamics of modulated waves on graphene like quantum graphs. (English) Zbl 1529.35466
This paper gives the long time dynamics of wave-packet solutions of the cubic Klein-Gordon equation set on infinite 2D periodic graphs. Examples of such 2D graphs are the honeycomb (Graphene) structure and the triangular structure. The graph periodicity is the fast scale of the wave-packet.
Because the linear analysis of the Klein Gordon equation on the honeycomb graph reduces to a 2D band-gap diagram looking very much like Graphen’s (with Dirac points) this work is a counter part on graphs of C. L. Fefferman and M. I. Weinstein’s results in \(\mathbb{R}^2\) (cf. [Commun. Math. Phys. 326, No. 1, 251–286 (2014; Zbl 1292.35195); Journ. Équ. Dériv. Partielles, St.-Jean-de-Monts 2012, Paper No. 12, 12 p. (2012; doi:10.5802/jedp.95)]).
The main differences are:
Because the linear analysis of the Klein Gordon equation on the honeycomb graph reduces to a 2D band-gap diagram looking very much like Graphen’s (with Dirac points) this work is a counter part on graphs of C. L. Fefferman and M. I. Weinstein’s results in \(\mathbb{R}^2\) (cf. [Commun. Math. Phys. 326, No. 1, 251–286 (2014; Zbl 1292.35195); Journ. Équ. Dériv. Partielles, St.-Jean-de-Monts 2012, Paper No. 12, 12 p. (2012; doi:10.5802/jedp.95)]).
The main differences are:
- ●
- One can get rid of the angles in the graph thus reducing for instance the honeycomb structure to a rectangular (brick) graph which allows to simplify the nonlinear analysis.
- ●
- Kirchhoff conditions at graph nodes require a higher order Ansatz in view of the convergence of the reduced model (NLS or Dirac) in \(L^2\)-Sobolev spaces.
Reviewer: Vincent Lescarret (Gif-sur-Yvette)
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 35Q40 | PDEs in connection with quantum mechanics |
| 35R02 | PDEs on graphs and networks (ramified or polygonal spaces) |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
Citations:
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