On the nonlinear Dirac equation on noncompact metric graphs

TL;DR

This paper investigates the nonlinear Dirac equation on noncompact metric graphs, establishing local well-posedness and the existence of bifurcating standing waves, while also exploring the nonrelativistic limit of the operator.

Contribution

It proves local well-posedness for the nonlinear Dirac equation on noncompact graphs and demonstrates the bifurcation of standing waves on infinite star graphs, extending understanding of relativistic quantum models on networks.

Findings

Proved local well-posedness in the operator domain.

Established existence of standing waves bifurcating from trivial solutions.

Analyzed the nonrelativistic limit of the Dirac-Kirchhoff operator.

Abstract

The paper discusses the Nonlinear Dirac Equation with Kerr-type nonlinearity (i.e., $\psi^{p-2}\psi$) on noncompact metric graphs with a finite number of edges, in the case of Kirchhoff-type vertex conditions. Precisely, we prove local well-posedness for the associated Cauchy problem in the operator domain and, for infinite $N$-star graphs, the existence of standing waves bifurcating from the trivial solution at $\omega=mc^2$, for any $p>2$. In the Appendix we also discuss the nonrelativistic limit of the Dirac-Kirchhoff operator.