On the nonlinear Dirac equation on noncompact metric graphs

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.