Nonrelativistic limit of bound-state solutions for nonlinear Dirac equation on noncompact quantum graphs

In this paper, we investigate the nonrelativistic limit and qualitative properties of bound-state solutions for the nonlinear Dirac equation (NLDE) defined on noncompact quantum graphs: $$-ic\frac{d}{dx}{\sigma }_1\psi +m{c}^2{\sigma }_3\psi -\omega \psi =g(\mathrm{\mid }\psi \mathrm{\mid })\psi ,\text{ in }\mathcal{G}$$ where $g:\mathbb{R}\to \mathbb{R}$ is a continuous nonlinear function, $c>0$ represents the speed of light, $m>0$ is the particle’s mass, $\omega \in \mathbb{R}$ is related to the frequency, ${\sigma }_1$ and ${\sigma }_3$ denote the Pauli matrices, and $\mathcal{G}$ is a noncompact quantum graph. We establish the existence of bound-state solutions to the NLDE on $\mathcal{G}$, and prove that these solutions converge toward the corresponding bound-state solutions of a nonlinear Schrödinger equation (NLS) in the nonrelativistic limit (i.e., as the speed of light $c\to \mathrm{\infty }$) for particles of small mass. Furthermore, we prove uniform boundedness and exponential decay properties of the NLDE solutions, uniformly in $c$, thereby offering insight into their asymptotic behavior.