Algebraic Origins of Reflectionless Scattering in Bogoliubov-de Gennes equations for a solitonic BEC

We consider small excitations around a one-dimensional bosonic soliton. It is well known that the corresponding Bogoliubov-de Gennes (BdG) Liouvillian features a vanishing reflection coefficient at all energies.\footnote{D. J. Kaup, J. Phys. A {\bf 42}, 5689 (1990).}$^,$\footnote{Y. Castin, Eur. Phys. J. B {\bf 68}, 317 (2009).} In this presentation, we show that this reflectionless property can be explained via an algebraic link (related to quantum-mechanical supersymmetry) to a potential-free Liouvillian.