2012/01/31 by Daisuke Takahashi, Daisuke A. Takahashi · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Cold Atom Physics and Bose-Einstein Condensates #Geometry #Hamiltonian (control theory) #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Photonic Systems #Nonlinear Schrödinger equation #Nonlinear system #Physics #Quantum mechanics #Saddle point #Scaling #Scattering #cond-mat.quant-gas #nlin.PS
paper · pdf · doi:10.1016/j.physd.2012.06.006
published as Physica D 241 (2012) 1589-1602 · 16 pages, 5 figures, elsarticle.cls, final version published in Physica D
openalex publication_date 2012/06/21 · arxiv created 2012/08/13 · arxiv updated 2012/08/14 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
A scattering problem (or more precisely, a transmission-reflection problem) of linearized excitations in the presence of a dark soliton is considered in a one-dimensional nonlinear Schrödinger system with a general nonlinearity: i∂t ϕ= -∂x2 ϕ+ F(|ϕ|2)ϕ. If the system is interpreted as a Bose-Einstein condensate, the linearized excitation is a Bogoliubov phonon, and the linearized equation is the Bogoliubov equation. We exactly prove that the perfect transmission of the zero-energy phonon is suppressed at a critical state determined by Barashenkov's stability criterion [Phys. Rev. Lett. 77, (1996) 1193.], and near the critical state, the energy-dependence of the reflection coefficient shows a saddle-node type scaling law. The analytical results are well supported by numerical calculation for cubic-quintic nonlinearity. Our result gives an exact example of scaling laws of saddle-node bifurcation in time-reversible Hamiltonian systems. As a by-product of the proof, we also give all exact zero-energy solutions of the Bogoliubov equation and their finite energy extension.