Normalized solutions of nonlinear Dirac equations on noncompact metric graphs with localized nonlinearities

TL;DR

This work addresses the existence of normalized bound states for nonlinear Dirac equations on noncompact metric graphs with nonlinearities localized to the compact core. It develops a graph-adapted variational framework, combining a Gagliardo-Nirenberg-Sobolev inequality on graphs with a penalized reduction scheme and mountain-pass geometry to obtain normalized solutions across subcritical ($2<p<4$), critical ($p=4$), and supercritical ($p>4$) regimes, under topological graph assumptions and in the presence of both positive and negative nonlinearities. The authors also treat the spectral edge case $\lambda=-mc^2$ as an eigenvalue and extend results to graphs augmented by long edges, yielding constructive existence results via topology (trees, cycles) and length parameters, as well as nonexistence results that sharpen the regimes of applicability. An appendix analyzes the dependence on physical parameters $(m,c>0)$, clarifying how these affect the existence landscape. Overall, the paper provides the first systematic treatment of normalized NLDE solutions on metric graphs, linking graph topology and spectral properties to variational existence results with broad implications for quantum graphs and nonlinear wave propagation.

Abstract

In this paper, we study the following nonlinear Dirac equations (NLDE) on noncompact metric graph $\mathcal{G}$ with localized nonlinearities \begin{equation} \mathcal{D} u - ωu= aχ_{\mathcal{K}}|u|^{p-2}u, \end{equation} where $\mathcal{D}$ is the Dirac operator on $\mathcal{G}$, $u: \mathcal{G} \to \mathbb{C}^2$, $ω\in \mathbb{R}$, $a > 0$, $χ_{\mathcal{K}}$ is the characteristic function of the compact core $\mathcal{K}$, and $p>2$. First, for $2<p<4$, we prove the existence of normalized solutions to (NLDE) using a perturbation argument. Then, for $p \geq 4$, we establish the assumption under which normalized solutions to (NLDE) exist. Finally, we extend these results to the case $a<0$ and, for all $p>2$, prove the existence of normalized solutions to (NLDE) when $λ= -mc^2$ is an eigenvalue of the operator $\mathcal{D}$. In the Appendix, we study the influence of the parameters $m, c > 0$ on the existence of normalized solutions to (NLDE). To the best of our knowledge, this is the first study to investigate the normalized solutions to (NLDE) on metric graphs.

Figures (2)

• Figure 1: A noncompact graph $\mathcal{G}$ with finitely number of edges and a non-empty compact core.
• Figure 2: $\mathcal{K}$ is a tree with at most one leaf incident with no half-line in $\mathcal{G}$

Theorems & Definitions (42)

• Definition 1.1: Bo
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Corollary 1.8
• Remark 1.9
• Lemma 2.1