Nonrelativistic limit of bound-state solutions for nonlinear Dirac equation on noncompact quantum graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917037422411776 |
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| author | Gu, Guangze Ruzhansky, Michael Wei, Guoyan Yang, Zhipeng |
| author_facet | Gu, Guangze Ruzhansky, Michael Wei, Guoyan Yang, Zhipeng |
| contents | 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: \[ -i c \frac{d}{d x} σ_1 ψ+m c^2 σ_3 ψ-ωψ=g(|ψ|) ψ, \quad \text { in } \mathcal{G} \] 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, \( ω\in\mathbb{R} \) is related to the frequency, \( σ_1 \) and \( σ_3 \) 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 \( c \to \infty \)) 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_20658 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonrelativistic limit of bound-state solutions for nonlinear Dirac equation on noncompact quantum graphs Gu, Guangze Ruzhansky, Michael Wei, Guoyan Yang, Zhipeng Analysis of PDEs Mathematical Physics 35R02, 35Q41, 81Q35 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: \[ -i c \frac{d}{d x} σ_1 ψ+m c^2 σ_3 ψ-ωψ=g(|ψ|) ψ, \quad \text { in } \mathcal{G} \] 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, \( ω\in\mathbb{R} \) is related to the frequency, \( σ_1 \) and \( σ_3 \) 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 \( c \to \infty \)) 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. |
| title | Nonrelativistic limit of bound-state solutions for nonlinear Dirac equation on noncompact quantum graphs |
| topic | Analysis of PDEs Mathematical Physics 35R02, 35Q41, 81Q35 |
| url | https://arxiv.org/abs/2510.20658 |