Mathematical results for the nonlinear Winter's model

In recent years, Winter's nonlinear model has been adopted in theoretical physics as the prototype for the study of quantum resonances and the dynamics of observables in the context of nonlinear Schr\"odinger equations. However, its mathematical treatment still has several important gaps. This article demonstrates a dispersive estimate of the evolution operator, from which the result of local well-posedeness of the solution follows; a criterion for the existence of the blow-up phenomenon is also provided. Finally, the phenomenon of bifurcations of stationary solutions is analysed, concluding with a conjecture on the orbital stability of some of them.