  The non-linear Hall term present in the induction equation in the
electron-magneto-hydrodynamics limit is responsible for the Hall drift of the
magnetic field and, in some cases, for the so-called Hall instability. We
investigate whether or not the growth rates and eigenfunctions found in the
linear analysis are consistent with the results of non-linear numerical
simulations. Following the linear analysis of Rheinhardt & Geppert, we study
the same cases for which the Hall instability was predicted by solving the
non-linear Hall induction equation using a two-dimensional conservative and
divergence-free finite difference scheme that overcomes intrinsic difficulties
of pseudo-spectral methods and can describe situations with arbitrarily high
magnetic Reynolds numbers. We show that unstable modes can grow to the level of
the background field without being overwhelmed by the Hall cascade, and cause a
complete rearrangement of the field geometry. We confirm both the growth rates
and eigenfunctions found in the linearized analysis and hence the instability.
In the non-linear regime, after the unstable modes grow to the background
level, the naturally selected modes become stable and oscillatory. Later on,
the evolution tends to select the modes with the longest possible wavelengths,
but this process occurs on the magnetic diffusion timescale. We confirm the
existence of the Hall instability. We argue against using the misleading
terminology that associates the non-linear Hall term with a turbulent Hall
cascade, since small-scale structures are not created everywhere. The field
evolves instead in a Burgers-like manner, forms local structures with strong
gradients which become shocks in the zero resistivity limit, and Hall waves are
launched and propagated through the entire domain.
