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 $\omega =m{c}^2$, for any $p>2$. In the Appendix we also discuss the nonrelativistic limit of the Dirac-Kirchhoff operator.

27 pages, 4 figures. Keywords: nonlinear Dirac equation, metric graphs, local well-posedness, bound states, implicit function theorem, bifurcation, perturbation method, nonrelativistic limit. The last subsection of the Appendix have been removed and some minor revisions have been made with respect to the previous version