Nonlinear Dirac equations on noncompact quantum graphs with potentials: Multiplicity and Concentration
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912704637173760 |
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| author | Gu, Guangze Li, Ziwei Ruzhansky, Michael Yang, Zhipeng |
| author_facet | Gu, Guangze Li, Ziwei Ruzhansky, Michael Yang, Zhipeng |
| contents | In this paper, we study the existence and multiplicity of solutions to the following class of nonlinear Dirac equations (NLDE) on noncompact quantum graphs:
\[
-i\,\varepsilon c\,σ_1\,\partial_x u + m c^2 σ_3 u + V(x)\,u = f(|u|)\,u, \quad x\in \mathcal{G}, \tag{P}
\]
where \(V:\mathcal{G}\to\mathbb{R}\) and \(f:\mathbb{R}\to\mathbb{R}\) are continuous, \(\varepsilon>0\) is a semiclassical parameter, \(m>0\) denotes the mass, and \(c>0\) the speed of light. Here \(σ_1,σ_3\) are Pauli matrices, and \(\mathcal{G}\) is a noncompact quantum graph. We prove that when \(\varepsilon\) is sufficiently small, the number of solutions to \((P)\) is at least the number of global minima of \(V\).
Moreover, these solutions exhibit semiclassical concentration: as \(\varepsilon\to0\), their concentration points approach the set of global minima of \(V\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_09285 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonlinear Dirac equations on noncompact quantum graphs with potentials: Multiplicity and Concentration Gu, Guangze Li, Ziwei Ruzhansky, Michael Yang, Zhipeng Analysis of PDEs Mathematical Physics 35R02, 35Q41, 81Q35 In this paper, we study the existence and multiplicity of solutions to the following class of nonlinear Dirac equations (NLDE) on noncompact quantum graphs: \[ -i\,\varepsilon c\,σ_1\,\partial_x u + m c^2 σ_3 u + V(x)\,u = f(|u|)\,u, \quad x\in \mathcal{G}, \tag{P} \] where \(V:\mathcal{G}\to\mathbb{R}\) and \(f:\mathbb{R}\to\mathbb{R}\) are continuous, \(\varepsilon>0\) is a semiclassical parameter, \(m>0\) denotes the mass, and \(c>0\) the speed of light. Here \(σ_1,σ_3\) are Pauli matrices, and \(\mathcal{G}\) is a noncompact quantum graph. We prove that when \(\varepsilon\) is sufficiently small, the number of solutions to \((P)\) is at least the number of global minima of \(V\). Moreover, these solutions exhibit semiclassical concentration: as \(\varepsilon\to0\), their concentration points approach the set of global minima of \(V\). |
| title | Nonlinear Dirac equations on noncompact quantum graphs with potentials: Multiplicity and Concentration |
| topic | Analysis of PDEs Mathematical Physics 35R02, 35Q41, 81Q35 |
| url | https://arxiv.org/abs/2511.09285 |