  In this article, we prove various illposedness results for the Cauchy problem
for the incompressible Hall- and electron-magnetohydrodynamic (MHD) equations
without resistivity. These PDEs are fluid descriptions of plasmas, where the
effect of collisions is neglected (no resistivity), while the motion of the
electrons relative to the ions (Hall current term) is taken into account. The
Hall current term endows the magnetic field equation with a quasilinear
dispersive character, which is key to our mechanism for illposedness. Perhaps
the most striking conclusion of this article is that the Cauchy problems for
the Hall-MHD (either viscous or inviscid) and the electron-MHD equations, under
one translational symmetry, are ill-posed near the trivial solution in any
sufficiently high regularity Sobolev space and even in Gevrey spaces. This
result holds despite obvious wellposedness of the linearized equations near the
trivial solution, as well as conservation of the nonlinear energy, by which the
$L^{2}$ norm (energy) of the solution stays constant in time. The core
illposedness mechanism is degeneration of certain high frequency wave packet
solutions to the linearization around a class of linearly degenerate stationary
solutions of these equations, which are essentially dispersive equations with
degenerate principal symbols. The method developed in this work is sharp and
robust, in that we also prove nonlinear $H^{s}$-illposedness in the presence of
fractional dissipation of any order less than 1, matching known wellposedness
results. The results in this article are complemented by a companion work,
where we provide geometric conditions on the initial magnetic field that ensure
wellposedness(!) of the Cauchy problem. In particular, in stark contrast to the
results here, it is shown in the companion work that the nonlinear Cauchy
problems are well-posed near any nonzero constant magnetic field.
