Oscillatory and regularized shock waves for a modified Serre-Green-Naghdi system
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911763008585728 |
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| author | Bolbot, Daria Mitsotakis, Dimitrios Tzavaras, Athanasios E. |
| author_facet | Bolbot, Daria Mitsotakis, Dimitrios Tzavaras, Athanasios E. |
| contents | The Serre-Green-Naghdi equations of water wave theory have been widely employed to study undular bores. In this study, we introduce a modified Serre-Green-Naghdi system incorporating the effect of an artificial term that results in dispersive and dissipative dynamics. We show that, over sufficiently extended time intervals, effectively approximates the classical Serre-Green-Naghdi equations and admits dispersive-diffusive shock waves as traveling wave solutions. The traveling waves converge to the entropic shock wave solution of the shallow water equations when the dispersion and diffusion approach zero in a moderate dispersion regime. These findings contribute to an understanding of the formation of dispersive shock waves in the classical Serre-Green-Naghdi equations and the effects of diffusion in the generation and propagation of undular bores. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_12411 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Oscillatory and regularized shock waves for a modified Serre-Green-Naghdi system Bolbot, Daria Mitsotakis, Dimitrios Tzavaras, Athanasios E. Analysis of PDEs The Serre-Green-Naghdi equations of water wave theory have been widely employed to study undular bores. In this study, we introduce a modified Serre-Green-Naghdi system incorporating the effect of an artificial term that results in dispersive and dissipative dynamics. We show that, over sufficiently extended time intervals, effectively approximates the classical Serre-Green-Naghdi equations and admits dispersive-diffusive shock waves as traveling wave solutions. The traveling waves converge to the entropic shock wave solution of the shallow water equations when the dispersion and diffusion approach zero in a moderate dispersion regime. These findings contribute to an understanding of the formation of dispersive shock waves in the classical Serre-Green-Naghdi equations and the effects of diffusion in the generation and propagation of undular bores. |
| title | Oscillatory and regularized shock waves for a modified Serre-Green-Naghdi system |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2401.12411 |