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Remarks on sine-Gordon kink--fermion system: localized modes and scattering

We study numerically the kink-fermion interactions in a 1+1 dimensional toy model, which describes sine-Gordon kinks coupled to the massless Dirac fermions with backreaction. We show that the spectrum of fermionic modes strongly depends on the choice of the coupling, in particular, there are no localized modes for a minimal Yukawa coupling. We analyze the scattering of the fermionic packet by the kink. We demonstrate that the outcome of the collision dynamically depends on the phase of the incoming fermion packet, it results in alternating regimes of positive and negative acceleration of the kink.