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

Guangze Gu; Michael Ruzhansky; Guoyan Wei; Zhipeng Yang

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arXiv · 2510.20658

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

Guangze Gu·Michael Ruzhansky·Guoyan Wei·Zhipeng Yang

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Abstract

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: \[ -i c \frac{d}{d x} \sigma_1 \psi+m c^2 \sigma_3 \psi-\omega \psi=g(|\psi|) \psi, \quad \text { in } \mathcal{G} \] where $ g : \mathbb{R}\rightarrow\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 Schr\"odinger equation (NLS) in the nonrelativistic limit (i.e., as the speed of light $ c \to \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.

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Guangze Gu, Michael Ruzhansky, Guoyan Wei, Zhipeng Yang. 2025-10-23. Nonrelativistic limit of bound-state solutions for nonlinear Dirac equation on noncompact quantum graphs. https://arxiv.org/abs/2510.20658

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