### Abstract

Lie-Bäcklund symmetries and conservation laws are derived for weakly nonlinear magnetohydrodynamic (MHD) equations describing the interaction of the Alfvén and magnetoacoustic modes propagating parallel to the ambient magnetic field, in the parameter regime near the triple umbilic point, where the gas sound speed a_{g} matches the Alfvén speed V_{A}. The dispersive form of the equations can be expressed in Hamiltonian form and admit four Lie point symmetries and conservation laws associated with space-translation invariance (momentum conservation), time translation invariance (energy conservation), rotational invariance about the magnetic field B (helicity conservation), plus a further symmetry that is associated with accelerating wave similarity solutions of the equations. The main aim of the paper is a study of the symmetries and conservation laws of the dispersionless equations. The dispersionless equations are of hydrodynamic type and have three families of characteristics analogous to the slow, intermediate and fast modes of MHD and the Riemann invariants for each of these modes are given in closed form. The dispersionless equations are shown to be semi-Hamiltonian, and to possess two infinite families of symmetries and conservation laws. The analysis emphasizes the role of the Riemann invariants of the dispersionless equations and a hodograph transformation for a restricted version of the equations.

Original language | English (US) |
---|---|

Pages (from-to) | 5209-5240 |

Number of pages | 32 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 29 |

Issue number | 16 |

DOIs | |

State | Published - 1996 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Physics A: Mathematical and General*,

*29*(16), 5209-5240. https://doi.org/10.1088/0305-4470/29/16/037

**Lie-Bäcklund symmetries of dispersionless, magnetohydrodynamic model equations near the triple umbilic point.** / Webb, G. M.; Brio, Moysey; Zank, G. P.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and General*, vol. 29, no. 16, pp. 5209-5240. https://doi.org/10.1088/0305-4470/29/16/037

}

TY - JOUR

T1 - Lie-Bäcklund symmetries of dispersionless, magnetohydrodynamic model equations near the triple umbilic point

AU - Webb, G. M.

AU - Brio, Moysey

AU - Zank, G. P.

PY - 1996

Y1 - 1996

N2 - Lie-Bäcklund symmetries and conservation laws are derived for weakly nonlinear magnetohydrodynamic (MHD) equations describing the interaction of the Alfvén and magnetoacoustic modes propagating parallel to the ambient magnetic field, in the parameter regime near the triple umbilic point, where the gas sound speed ag matches the Alfvén speed VA. The dispersive form of the equations can be expressed in Hamiltonian form and admit four Lie point symmetries and conservation laws associated with space-translation invariance (momentum conservation), time translation invariance (energy conservation), rotational invariance about the magnetic field B (helicity conservation), plus a further symmetry that is associated with accelerating wave similarity solutions of the equations. The main aim of the paper is a study of the symmetries and conservation laws of the dispersionless equations. The dispersionless equations are of hydrodynamic type and have three families of characteristics analogous to the slow, intermediate and fast modes of MHD and the Riemann invariants for each of these modes are given in closed form. The dispersionless equations are shown to be semi-Hamiltonian, and to possess two infinite families of symmetries and conservation laws. The analysis emphasizes the role of the Riemann invariants of the dispersionless equations and a hodograph transformation for a restricted version of the equations.

AB - Lie-Bäcklund symmetries and conservation laws are derived for weakly nonlinear magnetohydrodynamic (MHD) equations describing the interaction of the Alfvén and magnetoacoustic modes propagating parallel to the ambient magnetic field, in the parameter regime near the triple umbilic point, where the gas sound speed ag matches the Alfvén speed VA. The dispersive form of the equations can be expressed in Hamiltonian form and admit four Lie point symmetries and conservation laws associated with space-translation invariance (momentum conservation), time translation invariance (energy conservation), rotational invariance about the magnetic field B (helicity conservation), plus a further symmetry that is associated with accelerating wave similarity solutions of the equations. The main aim of the paper is a study of the symmetries and conservation laws of the dispersionless equations. The dispersionless equations are of hydrodynamic type and have three families of characteristics analogous to the slow, intermediate and fast modes of MHD and the Riemann invariants for each of these modes are given in closed form. The dispersionless equations are shown to be semi-Hamiltonian, and to possess two infinite families of symmetries and conservation laws. The analysis emphasizes the role of the Riemann invariants of the dispersionless equations and a hodograph transformation for a restricted version of the equations.

UR - http://www.scopus.com/inward/record.url?scp=22244483805&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=22244483805&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/29/16/037

DO - 10.1088/0305-4470/29/16/037

M3 - Article

AN - SCOPUS:22244483805

VL - 29

SP - 5209

EP - 5240

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 16

ER -