Lie Symmetries for the Shallow Water Magnetohydrodynamics Equations in a Rotating Reference Frame
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arXiv
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| Natura: | Preprint |
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2024
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| author | Paliathanasis, Andronikos Halder, Amlan |
| author_facet | Paliathanasis, Andronikos Halder, Amlan |
| contents | We perform a detailed Lie symmetry analysis for the hyperbolic system of partial differential equations that describe the one-dimensional Shallow Water magnetohydrodynamics equations within a rotating reference frame. We consider a relaxing condition $\mathbf{\mathbf{\nabla }}\left( h\mathbf{B} \right) \neq 0$ for the one-dimensional problem, which has been used to overcome unphysical behaviors. The hyperbolic system of partial differential equations depends on two parameters: the constant gravitational potential $g$ and the Coriolis term $f_{0}$, related to the constant rotation of the reference frame. For four different cases, namely $g=0,~f_{0}=0$; $g\neq 0\,,~f_{0}=0$; $g=0$, $f_{0}\neq 0$; and $g\neq 0$, $f_{0}\neq 0$ the admitted Lie symmetries for the hyperbolic system form different Lie algebras. Specifically the admitted Lie algebras are the $L^{10}=\left\{ A_{3,3}\rtimes A_{2,1}\right\} \otimes _{s}A_{5,34}^{a}$; $% L^{8}=A_{2,1}\rtimes A_{6,22}$; $L^{7}=A_{3,5}\rtimes\left\{ A_{2,1}\rtimes A_{2,1}\right\} $; and $L^{6}=A_{3,5}\rtimes A_{3,3}~$respectively, where we use the Morozov-Mubarakzyanov-Patera classification scheme. For the general case where $f_{0}g\neq 0$, we derive all the invariants for the Adjoint action of the Lie algebra $L^{6}$ and its subalgebras, and we calculate all the elements of the one-dimensional optimal system. These elements are then considered to define similarity transformations and construct analytic solutions for the hyperbolic system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_14578 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lie Symmetries for the Shallow Water Magnetohydrodynamics Equations in a Rotating Reference Frame Paliathanasis, Andronikos Halder, Amlan Mathematical Physics Analysis of PDEs Plasma Physics We perform a detailed Lie symmetry analysis for the hyperbolic system of partial differential equations that describe the one-dimensional Shallow Water magnetohydrodynamics equations within a rotating reference frame. We consider a relaxing condition $\mathbf{\mathbf{\nabla }}\left( h\mathbf{B} \right) \neq 0$ for the one-dimensional problem, which has been used to overcome unphysical behaviors. The hyperbolic system of partial differential equations depends on two parameters: the constant gravitational potential $g$ and the Coriolis term $f_{0}$, related to the constant rotation of the reference frame. For four different cases, namely $g=0,~f_{0}=0$; $g\neq 0\,,~f_{0}=0$; $g=0$, $f_{0}\neq 0$; and $g\neq 0$, $f_{0}\neq 0$ the admitted Lie symmetries for the hyperbolic system form different Lie algebras. Specifically the admitted Lie algebras are the $L^{10}=\left\{ A_{3,3}\rtimes A_{2,1}\right\} \otimes _{s}A_{5,34}^{a}$; $% L^{8}=A_{2,1}\rtimes A_{6,22}$; $L^{7}=A_{3,5}\rtimes\left\{ A_{2,1}\rtimes A_{2,1}\right\} $; and $L^{6}=A_{3,5}\rtimes A_{3,3}~$respectively, where we use the Morozov-Mubarakzyanov-Patera classification scheme. For the general case where $f_{0}g\neq 0$, we derive all the invariants for the Adjoint action of the Lie algebra $L^{6}$ and its subalgebras, and we calculate all the elements of the one-dimensional optimal system. These elements are then considered to define similarity transformations and construct analytic solutions for the hyperbolic system. |
| title | Lie Symmetries for the Shallow Water Magnetohydrodynamics Equations in a Rotating Reference Frame |
| topic | Mathematical Physics Analysis of PDEs Plasma Physics |
| url | https://arxiv.org/abs/2412.14578 |