Shocks and discontinuities in the Hall-MHD model

There are a number of papers already published in the literature which present fundamentally incorrect results on shocks and other discontinuities possible within the Hall-MHD model. The basic mistake is that the previous works assume that the jump conditions across a discontinuity must be local, connecting values of the field in the same location on both sides of the discontinuity. But this is incorrect. For example, ``massless'' electrons can travel along the discontinuity surface at ``infinite'' speed, enter the surface in one location and exit at another. We work out the shock-structure problem and present other explicit examples for the rather small number of discontinuities allowed within this model. As the shock strength diminishes, we show that the results converge to our recently dereived [1] dispersion relations of Hall-MHD waves. [1] E. Hameiri, A. Ishizawa, and A. Ishida, Phys. Plasmas\textbf{ 12, }072109 (2005).