Fermions on the kink revisited

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.