  The kinetic theory for the instabilities driven by the Hall current with a
sheared current velocity, which has the method of the shearing modes or the
so-called non-modal approach as its foundation, is developed. The developed
theory predicts that in the Hall plasma with the inhomogeneous electric field,
the separate spatial Fourier mode of the perturbations is determined in the
frame convected with one of the plasma components. Because of the different
shearing of the ion and electron flows in the Hall plasma, this mode is
perceived by the second component as the Doppler-shifted continuously sheared
mode with time-dependent wave numbers. Due to this effect, the interaction of
the plasma components forms the nonmodal time-dependent process, which should
be investigated as the initial value problem. The developed approach is applied
to the solutions of the linear initial value problems for the hydrodynamic
modified two-stream instability and the kinetic ion-sound instability of the
plasma with a sheared Hall current with a uniform velocity shear. These
solutions reveal that the uniform part of the current velocity is responsible
for the modal evolution of the instability, whereas the current velocity shear
is the source of the development of the nonmodal instability with exponent
growing with time as $\sim\left(t-t_0\right)^3$.
