Lax representations for the magnetohydrodynamics equations
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866916589283049472 |
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| author | Morozov, Oleg I. |
| author_facet | Morozov, Oleg I. |
| contents | We find two Lax representations for the reduced magnetohydrodynamics equations ({\sc rmhd}) and derive a B{ä}cklund transformation between the tangent and the cotangent coverings of these equations. Then, we study the action of the B{ä}cklund transformation on the second-order cosymmetries and the action of its inverse on the Lie symmetries of {\sc rmhd}. The action of the inverse transformation produces another Lax representation for {\sc rmhd}. The reduction of {\sc rmhd} by the symmetry of shifts along the $z$-axis coincides with the equations of two-dimensional ideal magnetohydrodynamics ({\sc imhd}). Applied to the Lax representations and the B{ä}cklund transformation of {\sc rmhd}, the reduction provides analogous constructions for {\sc imhd}. The action of the inverse B{ä}cklund transformation on the Lie symmetries of {\sc imhd} is expressed in terms of a new four-parameter Lax representation of these equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_07411 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lax representations for the magnetohydrodynamics equations Morozov, Oleg I. Exactly Solvable and Integrable Systems We find two Lax representations for the reduced magnetohydrodynamics equations ({\sc rmhd}) and derive a B{ä}cklund transformation between the tangent and the cotangent coverings of these equations. Then, we study the action of the B{ä}cklund transformation on the second-order cosymmetries and the action of its inverse on the Lie symmetries of {\sc rmhd}. The action of the inverse transformation produces another Lax representation for {\sc rmhd}. The reduction of {\sc rmhd} by the symmetry of shifts along the $z$-axis coincides with the equations of two-dimensional ideal magnetohydrodynamics ({\sc imhd}). Applied to the Lax representations and the B{ä}cklund transformation of {\sc rmhd}, the reduction provides analogous constructions for {\sc imhd}. The action of the inverse B{ä}cklund transformation on the Lie symmetries of {\sc imhd} is expressed in terms of a new four-parameter Lax representation of these equations. |
| title | Lax representations for the magnetohydrodynamics equations |
| topic | Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2303.07411 |