  We study fermion modes localized on the kink in the 1+1 dimensional $\phi^4$
model, coupled to the Dirac fermions with backreaction. Using numerical methods
we construct self-consistent solutions of the corresponding system of coupled
integral-differential equations and study dependencies of the scalar field of
the kink and the normalizable fermion bound states on the values of the values
of the parameters of the model. We show that the backreaction of the localized
fermions significantly modifies the solutions, in particular it results in
spatial oscillations of the profile of the kink and violations of the
reflection symmetry of the configuration.
