On reflectionless nature of self-consistent multi-soliton solutions in Bogoliubov-de gennes and chiral Gross-Neveu models

Recently the most general static self-consistent multi-soliton solutions in Bogoliubov-de Gennes and chiral Gross-Neveu systems were derived by the present authors (Takahashi and Nitta, Phys. Rev. Lett. 110:131601, 2013). Here we show a few complementary results, which were absent in our previous work. We prove directly from the gap equation that the self-consistent solutions need to have reflectionless potentials. We also give the self-consistent condition for the system consisting of only right-movers, which is more used in high-energy physics.