Exhaustive derivation of static self-consistent multi-soliton solutions in the matrix Bogoliubov-de Gennes systems

The matrix-generalized Bogoliubov-de Gennes systems have recently been considered by the present author [arXiv:1509.04242, Phys. Rev. B 93, 024512 (2016)], and time-dependent and self-consistent multi-soliton solutions have been constructed based on the ansatz method. In this paper, restricting the problem to the static case, we exhaustively determine the self-consistent solutions using the inverse scattering theory. Solving the gap equation, we rigorously prove that the self-consistent potential must be reflectionless. As a supplementary topic, we elucidate the relation between the stationary self-consistent potentials and the soliton solutions in the matrix nonlinear Schr\"odinger equation. Asymptotic formulae of multi-soliton solutions for sufficiently isolated solitons are also presented.