Existence of undercompressive travelling waves of a non-local generalised Korteweg-de Vries-Burgers equation
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| author | Achleitner, F. Cuesta, C. M. Diez-Izagirre, X. |
| author_facet | Achleitner, F. Cuesta, C. M. Diez-Izagirre, X. |
| contents | We study travelling wave solutions of a generalised Korteweg-de Vries-Burgers equation with a non-local diffusion term and a concave-convex flux. This model equation arises in the analysis of a shallow water flow by performing formal asymptotic expansions associated to the triple-deck regularisation (which is an extension of classical boundary layer theory). The resulting non-local operator is a fractional type derivative with order between $1$ and $2$. Travelling wave solutions are typically analysed in relation to shock formation in the full shallow water problem. We show rigorously the existence of travelling waves that, formally, in the limit of vanishing diffusion and dispersion would give rise to non-classical shocks, that is, shocks that violate the Lax entropy condition. The proof is based on arguments that are typical in dynamical systems. The nature of the non-local operator makes this possible, since the resulting travelling wave equation can be seen as a delayed integro-differential equation. Thus, linearisation around critical points, continuity with respect to parameters and a shooting argument, are the main steps that we have proved and adapted for solving this problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_03209 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence of undercompressive travelling waves of a non-local generalised Korteweg-de Vries-Burgers equation Achleitner, F. Cuesta, C. M. Diez-Izagirre, X. Analysis of PDEs 47J35, 26A33, 35C07 We study travelling wave solutions of a generalised Korteweg-de Vries-Burgers equation with a non-local diffusion term and a concave-convex flux. This model equation arises in the analysis of a shallow water flow by performing formal asymptotic expansions associated to the triple-deck regularisation (which is an extension of classical boundary layer theory). The resulting non-local operator is a fractional type derivative with order between $1$ and $2$. Travelling wave solutions are typically analysed in relation to shock formation in the full shallow water problem. We show rigorously the existence of travelling waves that, formally, in the limit of vanishing diffusion and dispersion would give rise to non-classical shocks, that is, shocks that violate the Lax entropy condition. The proof is based on arguments that are typical in dynamical systems. The nature of the non-local operator makes this possible, since the resulting travelling wave equation can be seen as a delayed integro-differential equation. Thus, linearisation around critical points, continuity with respect to parameters and a shooting argument, are the main steps that we have proved and adapted for solving this problem. |
| title | Existence of undercompressive travelling waves of a non-local generalised Korteweg-de Vries-Burgers equation |
| topic | Analysis of PDEs 47J35, 26A33, 35C07 |
| url | https://arxiv.org/abs/2412.03209 |