Nonlinear Dirac Equation on Graphs with Localized Nonlinearities: Bound States and Nonrelativistic Limit

Abstract

In this paper we study the nonlinear Dirac (NLD) equation on noncompact metric graphs with localized Kerr nonlinearities, in the case of Kirchhoff-type conditions at the vertices. Precisely, we discuss existence and multiplicity of the bound states (arising as critical points of the NLD action functional) and we prove that, in the L 2 -subcritical case, they converge to the bound states of the NLS equation in the nonrelativistic limit. 1The attention recently attracted by the linear and the nonlinear Dirac equations is due to their applications, as effective equations, in many physical models, as in solid state physics and nonlinear optics [33,34].While originally the NLDE appeared as a field equation for relativistic interacting fermions [38], then it was used in particle physics to simulate features of quark confinement, acoustic physics, and in the context of Bose-Einstein condensates [34].Recently, it also appeared that some properties of physical models, as thin carbon structures, are well described using as an effective equation for non-relativistic electronic properties , the Dirac equation. We mention, thereupon, the seminal papers by Fefferman and Weinstein [28,29], the work of Arbunich and Sparber [10] (where a rigorous justification of linear and nonlinear equations in two-dimensional honeycomb structures is given) and the referenced therein. In addition, we

Quantum graphs and dimensional crossover: the honeycomb

“…p = 6 [6]). More recently, also some pioneering investigations of nonlinear Dirac equations has been initiated ( [9,10]).…”

“…The remainder of the paper is organised as follows. Section 2 sets some notation for the honeycomb, whereas Section 3 develops the proof of Sobolev inequality (9). Finally, within Section 4 we exhibit functions realizing strictly negative energy when p ∈ (2, 4), giving the proof of Theorem 1.1.…”

An Overview on the Standing Waves of Nonlinear Schrödinger and Dirac Equations on Metric Graphs with Localized Nonlinearity

“…Recently, we started a new research project concerning the NonLinear Dirac Equation (NLDE) on metric graphs in [18]. The physical motivations for such a model mainly come from solid state physics and nonlinear optics.…”

“…In fact, in order to deal with more complex graph topologies, in [18] we restricted ourselves to the case of a Kirchhoff-type extension of the Dirac operator (for details see Section 3.1) and, most importantly, we considered the case of a localized nonlinearity, that is…”

“…We expect that this will require some technical adjustments in order to deal with the indefiniteness of the nonlinearity. This might be quite relevant in the study of the nonrelativistic limit, since in [18] we used the fact that a positive nonlinearity allows to get a priori estimates uniform in c.…”