  We investigate a coupled system of a Dirac particle and a pseudoscalar field
in the form of a soliton in (1+1) dimensions and find some of its exact
solutions numerically.
  We solve the coupled set of equations self-consistently and
non-perturbatively by the use of a numerical method and obtain the bound states
of the fermion and the shape of the soliton. That is the shape of the static
soliton in this problem is not prescribed and is determined by the equations
themselves. This work goes beyond the perturbation theory in which the back
reaction of the fermion on soliton is its first order correction. We compare
our results to those of an exactly solvable model in which the soliton is
prescribed. We show that, as expected, the total energy of our system is lower
than the prescribed one. We also compute non-perturbatively the vacuum
polarization of the fermion induced by the presence of the soliton and display
the results. Moreover, we compute the soliton mass as a function of the boson
and fermion masses and find that the results are consistent with Skyrme's
phenomenological conjecture. Finally, we show that for fixed values of the
parameters, the shape of the soliton obtained from our exact solutions depends
slightly on the fermionic state to which it is coupled. However, the exact
shape of the soliton is always very close to the isolated kink.
