Barrier transmission for the one-dimensional nonlinear Schr\"odinger equation: resonances and transmission profiles

TL;DR

This paper investigates how nonlinear effects alter the transmission profiles in one-dimensional Schrödinger equations, linking these profiles to complex resonances and extending methods for calculating nonlinear resonances.

Contribution

It introduces a novel analysis of nonlinear transmission functions, relating them to Siegert resonances, and extends the Siegert method for nonlinear resonance calculations.

Findings

Nonlinear transmission profiles deviate from Lorentzian shapes.

Skeleton functions effectively describe nonlinear transmission and resonance decay.

Extended Siegert method facilitates calculation of nonlinear resonances.

Abstract

The stationary nonlinear Schr\"odinger equation (or Gross-Pitaevskii equation) for one-dimensional potential scattering is studied. The nonlinear transmission function shows a distorted profile, which differs from the Lorentzian one found in the linear case. This nonlinear profile function is analyzed and related to Siegert type complex resonances. It is shown, that the characteristic nonlinear profile function can be conveniently described in terms of skeleton functions depending on a few instructive parameters. These skeleton functions also determine the decay behavior of the underlying resonance state. Furthermore we extend the Siegert method for calculating resonances, which provides a convenient recipe for calculating nonlinear resonances. Applications to a double Gaussian barrier and a square well potential illustrate our analysis.