Multiple Riemann wave solutions of the general form of quasilinear hyperbolic systems
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866912710698991616 |
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| author | Grundland, A. M. de Lucas, J. |
| author_facet | Grundland, A. M. de Lucas, J. |
| contents | The objective of this paper is to construct geometrically Riemann $k$-wave solutions of the general form of first-order quasilinear hyperbolic systems of partial differential equations. To this end, we adapt and combine elements of two approaches to the construction of Riemann $k$-waves, namely the symmetry reduction method and the generalized method of characteristics. We formulate a geometrical setting for the general form of the $k$-wave problem and discuss in detail the conditions for the existence of $k$-wave solutions. An auxiliary result concerning the Frobenius theorem is established. We use it to obtain formulae describing the $k$-wave solutions in closed form. Our theoretical considerations are illustrated by examples of hydrodynamic type systems including the Brownian motion equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_15122 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Multiple Riemann wave solutions of the general form of quasilinear hyperbolic systems Grundland, A. M. de Lucas, J. Analysis of PDEs Mathematical Physics Exactly Solvable and Integrable Systems 35Q53 (primary), 35A30, 35Q58, 53A05 (secondary) The objective of this paper is to construct geometrically Riemann $k$-wave solutions of the general form of first-order quasilinear hyperbolic systems of partial differential equations. To this end, we adapt and combine elements of two approaches to the construction of Riemann $k$-waves, namely the symmetry reduction method and the generalized method of characteristics. We formulate a geometrical setting for the general form of the $k$-wave problem and discuss in detail the conditions for the existence of $k$-wave solutions. An auxiliary result concerning the Frobenius theorem is established. We use it to obtain formulae describing the $k$-wave solutions in closed form. Our theoretical considerations are illustrated by examples of hydrodynamic type systems including the Brownian motion equation. |
| title | Multiple Riemann wave solutions of the general form of quasilinear hyperbolic systems |
| topic | Analysis of PDEs Mathematical Physics Exactly Solvable and Integrable Systems 35Q53 (primary), 35A30, 35Q58, 53A05 (secondary) |
| url | https://arxiv.org/abs/2203.15122 |