Nonlinearity-defect interaction: symmetry breaking bifurcation in a NLS with a $\delta$' impurity

We illustrate some new results and comment on perspectives of a recent research line, focused on the stability of stationary states of nonlinear NLS with point interactions. We describe in detail the case of a "$\delta'$" interaction, that provides a rich model endowed with a pitchfork bifurcation with symmetry breaking in the family of ground states. Finally, we give a direct proof of the stability of the ground states in the cases of a subcritical and critical (in the sense of the blow-up) nonlinearity power.