Mathematical results for the nonlinear Winter's model

TL;DR

This paper provides new mathematical insights into Winter's nonlinear model, establishing dispersive estimates, criteria for blow-up, and analyzing bifurcations of stationary solutions in the context of nonlinear Schrödinger equations.

Contribution

It offers the first dispersive estimate for the evolution operator and explores bifurcation phenomena, filling key gaps in the mathematical understanding of Winter's model.

Findings

Dispersive estimate of the evolution operator established

Criterion for blow-up phenomenon provided

Analysis of bifurcations of stationary solutions conducted

Abstract

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.