Singular traveling waves for the Euler-Poisson system
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866910135509581824 |
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| author | Guelmame, Billel Hmidi, Taoufik Houamed, Haroune Rousset, Frédéric |
| author_facet | Guelmame, Billel Hmidi, Taoufik Houamed, Haroune Rousset, Frédéric |
| contents | We consider the Euler-Poisson system for ions where the electrons are given by a Maxwell-Boltzmann distribution, and we investigate the existence of one-dimensional periodic traveling waves. More precisely, we first establish the existence of a smooth global branch of bifurcation emanating from a constant equilibrium. We then construct a singular traveling wave emerging as the limiting profile at the end of the global curve of bifurcation.
Our analysis accommodates a wide class of pressure laws and provides a comprehensive characterization of both smooth and singular traveling waves. A central difficulty in this model arises from the exponential nonlinearity, induced by the nonlocal Poisson-Boltzmann equation, which prevents any explicit representation of the electron field in terms of the ion density. This poses significant obstacles compared to previous studies on related models, where such explicit formulas were crucial for global bifurcation arguments. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_14997 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Singular traveling waves for the Euler-Poisson system Guelmame, Billel Hmidi, Taoufik Houamed, Haroune Rousset, Frédéric Analysis of PDEs We consider the Euler-Poisson system for ions where the electrons are given by a Maxwell-Boltzmann distribution, and we investigate the existence of one-dimensional periodic traveling waves. More precisely, we first establish the existence of a smooth global branch of bifurcation emanating from a constant equilibrium. We then construct a singular traveling wave emerging as the limiting profile at the end of the global curve of bifurcation. Our analysis accommodates a wide class of pressure laws and provides a comprehensive characterization of both smooth and singular traveling waves. A central difficulty in this model arises from the exponential nonlinearity, induced by the nonlocal Poisson-Boltzmann equation, which prevents any explicit representation of the electron field in terms of the ion density. This poses significant obstacles compared to previous studies on related models, where such explicit formulas were crucial for global bifurcation arguments. |
| title | Singular traveling waves for the Euler-Poisson system |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.14997 |