Resonance solutions of the nonlinear Schr\"odinger equation in an open double-well potential

TL;DR

This paper investigates how nonlinearity affects resonance states and decay dynamics in the nonlinear Schrödinger equation within an open double-well potential, providing analytical solutions and stability analysis.

Contribution

It introduces a flexible double delta-shell potential model with analytical solutions, exploring bifurcations, stability, and a discrete biorthogonal basis approximation for nonlinear resonance analysis.

Findings

Nonlinearity significantly influences decay dynamics.

Bifurcation scenarios of resonance states are characterized.

A biorthogonal basis approach accurately describes the system.

Abstract

The resonance states and the decay dynamics of the nonlinear Schr\"odinger (or Gross-Pitaevskii) equation are studied for a simple, however flexible model system, the double delta-shell potential. This model allows analytical solutions and provides insight into the influence of the nonlinearity on the decay dynamics. The bifurcation scenario of the resonance states is discussed, as well as their dynamical stability properties. A discrete approximation using a biorthogonal basis is suggested which allows an accurate description even for only two basis states in terms of a nonlinear, nonhermitian matrix problem.