A method for determining many new magnetohydrodynamic shock normal expressions is described that uses both magnetic-field and plasma-flow parameters. These shock normal expressions are useful as a check on the usual magnetic coplanarity expression or for cases in which the most reliable data are mixed. For high Alfv,•n Mach numbers, the exact results are well approximated by the simple velocity coplanarity formula. This is illustrated by calculating shock normals for the 1965 bow shock crossing of Pioneer 6.
The propagation of small amplitude hydromagnetic waves of a collisionless, anisotropic plasma in a strong magnetic field is treated using the Chew—Goldberger—Low (CGL) fluid equations. The Alfvén, fast and slow magnetoacoustic, and entropy modes are analysed by phase speed and wave front diagrams and compared to their counterparts in magnetohydrodynamics (MHD) where collisions maintain an isotropic pressure. Two main classes of CGL waves are identified, pseudo-MHD and reverse-MHD where the latter has the slow wave speed greater than the Alfvén wave speed. Significant differences of the CGL hydromagnetic waves as compared to MHD waves are found which have special relevance for a study of infinitesimal and finite discontinuities in space plasmas.
An approach for determining a class of master partial differential equations from which Type II hidden point symmetries are inherited is presented. As an example a model nonlinear partial differential equation (PDE) reduced to a target PDE by a Lie symmetry gains a Lie point symmetry that is not inherited (hidden) from the original PDE. On the other hand this Type II hidden symmetry is inherited from one or more of the class of master PDEs. The class of master PDEs is determined by the hidden symmetry reverse method. The reverse method is extended to determine symmetries of the master PDEs that are not inherited. We indicate why such methods are necessary to determine the genesis of Type II symmetries of PDEs as opposed to those that arise in ordinary differential equations (ODEs).
The provenance of Type II hidden point symmetries of differential equations reduced from nonlinear partial differential equations is analyzed. The hidden symmetries are extra symmetries in addition to the inherited symmetries of the differential equations when the number of independent and dependent variables is reduced by a Lie point symmetry. These Type II hidden symmetries do not arise from contact symmetries or nonlocal symmetries as in the case of ordinary differential equations. The Lie point symmetries of a model equation and the two-dimensional Burgers' equation and their descendants are used to identify the hidden symmetries. The significant new result is the provenance of the Type II Lie point hidden symmetries found for differential equations reduced from partial differential equations. Two methods for determining the source of the hidden symmetries are developed.
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