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.

arXiv comment27 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

References in corpus (29)

• Dirac materials
• Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics
• The Nonlinear Dirac Equation in Bose-Einstein Condensates: Foundation and Symmetries
• Wave packets in Honeycomb Structures and Two-Dimensional Dirac Equations
• Spectral Statistics for the Dirac Operator on Graphs
• Negative energy ground states for the $L^2$-critical NLSE on metric graphs
• Fast solitons on star graphs
• Optical analogue of relativistic Dirac solitons in binary waveguide arrays
• Stationary scattering from a nonlinear network
• NLS ground states on metric graphs with localized nonlinearities
• Ground state and orbital stability for the NLS equation on a general starlike graph with potentials
• Bound states of the NLS equation on metric graphs with localized nonlinearities
• On the lack of bound states for certain NLS equations on metric graphs
• Bifurcations of standing localized waves on periodic graphs
• Stationary waves on nonlinear quantum graphs: General framework and canonical perturbation theory
• $L^2$-critical NLS on noncompact metric graphs with localized nonlinearity: topological and metric features
• Dimensional crossover with a continuum of critical exponents for NLS on doubly periodic metric graphs
• Dynamics of Dirac solitons in networks
• Nonlinear Dirac Equation On Graphs With Localized Nonlinearities: Bound States And Nonrelativistic Limit
• Ground states of the $L^2$-critical NLS equation with localized nonlinearity on a tadpole graph
• Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity
• Nonlinear dynamics on branched structures and networks
• Equilateral quantum graphs and boundary triples
• A note on the Dirac operator with Kirchoff-type vertex conditions on noncompact metric graphs
• Sharp decay estimates for critical Dirac equations
• Blow-up and strong instability of standing waves for the NLS-$\delta$ equation on a star graph
• Multiple solutions for a self-consistent Dirac equation in two dimensions
• An overview on the standing waves of nonlinear Schrödinger and Dirac equations on metric graphs with localized nonlinearity
• Controllability of localized quantum states on infinite graphs through bilinear control fields

Cited by in corpus (8)

• Standing waves on quantum graphs
• A note on the Dirac operator with Kirchoff-type vertex conditions on noncompact metric graphs
• Spectral properties of relativistic quantum waveguides
• Symmetry breaking in two-dimensional square grids: persistence and failure of the dimensional crossover
• Prescribed mass ground states for a doubly nonlinear Schrödinger equation in dimension one
• Soliton dynamics in the ABS nonlinear spinor model with external fields
• Low-regularity well-posedness for a mixed-sign quadratic Dirac equation on $N$-star metric graphs
• ${\mathsf{D}}^2=\mathsf{H}+1/4$ with point interactions