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

摘要

The stationary nonlinear Schr\"odinger equation (or Gross-Pitaevskiiequation) for one-dimensional potential scattering is studied. The nonlineartransmission function shows a distorted profile, which differs from theLorentzian one found in the linear case. This nonlinear profile function isanalyzed and related to Siegert type complex resonances. It is shown, that thecharacteristic nonlinear profile function can be conveniently described interms of skeleton functions depending on a few instructive parameters. Theseskeleton functions also determine the decay behavior of the underlyingresonance state. Furthermore we extend the Siegert method for calculatingresonances, which provides a convenient recipe for calculating nonlinearresonances. Applications to a double Gaussian barrier and a square wellpotential illustrate our analysis.