Fold Points and Singularities in Hall MHD Differential–Algebraic Equations

We consider the singularity crossing phenomenon in differential-algebraic equations (DAEs) of Hall MHD systems in one spatial dimension. The Hall MHD DAEs have singularities with impasse points, pseudoequilibrium points, or singularity-induced bifurcation (SIB) points. The pseudoequilibrium and SIB points allow for smooth transitions between the plus (supersonic) and minus (subsonic) Riemann sheets. Within the singular pseudoequilibrium points, there may exist only one analytic trajectory crossing the sonic curve (as in the case of SIB point), two analytic and two other trajectories of lower degree of smoothness in the case of pseudosaddle points, or two analytic and an uncountable number of trajectories of lower smoothness in the case of singular pseudonodes. In this paper, we show examples of singular points in Hall MHD systems described by DAEs and explain the singularity (also called the sonic or forbidden curve) crossing phenomenon by using the recent developments in the qualitative analysis of DAEs.