Mathematical results for the nonlinear Winter's model

TL;DR

The work analyzes a nonlinear Schrödinger equation on the half-line with a delta-shell barrier at $x=a$ and a Dirichlet boundary at $x=0$, formulated as $i\dot\psi_t = H_\alpha \psi_t + \eta |\psi_t|^{2\sigma}\psi_t$. It establishes a dispersive estimate for the linear evolution, enabling local well-posedness results and a variance-based blow-up criterion for the nonlinear problem. The paper then comprehensively studies stationary states for both focusing and defocusing cubic nonlinearities, deriving explicit profile forms via Jacobi elliptic functions and identifying bifurcation structures, including numerical saddle-node cascades. A conjecture on the orbital stability of some stationary states is proposed, highlighting the potential stability of certain nonlinear bound-like states in Winter-type systems.

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 Schrödinger 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.

Figures (1)

• Figure 1: For $a=1$ and $\alpha =-4$ then for $\eta \in [-110,0]$ two branches $\Omega_1^\pm (\eta )$ and $\Omega_2^\pm (\eta )$ of stationary solutions raise at the bifurcation points corresponding to the two circles. The stationary solution obtained for the linear problem admits the continuation $\Omega_0(\eta )$.

Theorems & Definitions (23)

• Proposition 1
• proof
• Remark 1
• Theorem 1
• Remark 2
• Remark 3
• proof
• Lemma 1
• proof
• Lemma 2