Physical and Algebraic Origins of the Reflectionless Property of the Bogoliubov-de-Gennes-sine-Gordon Equation Around a Soliton

We analyze the reflectionless property of the so-called Bogoliubov-de-Gennes-sine-Gordon (BdG-sG) equation---a Sine-Gordon equation that has been linearized around a single soliton solution. We demonstrate that the absence of reflection is necessary for the original nonlinear soliton be transparent for the small breathers. On the other hand, we show that the BdG-sG equation is equivalent to the P\"{o}schl-Teller (PT) potential at transparency, whose transparency, in turn, originates from a SUSY structure of the PT Hamiltonian. Our study provides yet another example of a connection between the Supersymmetric Quantum Mechanics\footnote{E. Witten, Nucl. Phys. B 188, 513 (1981)} and integrable partial differential equations, in addition to the known examples of the Lax operators for the Korteweg-de Vries, sine-Gordon, and Nonlinear Schr\"{o}dinger equations that are shown to a have a SUSY structure for some few-solitonic solutions.\footnote{C. V. Sukumar, 1986 J. Phys. A 19, 2297 (1986); A. Koller and M. Olshanii, Phys. Rev. E 84, 066601 (2011)}