Exact self-consistent soliton dynamics based on the Bogoliubov--de Gennes (BdG) formalism in unconventional Fermi superfluids/superconductors possessing an
SU(d)-symmetric two-body interaction is presented. The derivation is based on the ansatz having the similar form as the Gelfand-Levitan-Marchenko equation in the inverse scattering theory. Our solutions can be regarded as a multicomponent generalization of the solutions recently derived by Dunne and Thies [Phys. Rev. Lett. 111, 121602 (2013)]. We also propose superpositions of occupation states, which make it possible to realize various filling rates even in one-flavor systems, and include Dirac and Majorana fermions. The soliton solutions in the
d=2 systems, which describe the mixture of singlet
s-wave and triplet
p-wave superfluids, exhibit a variety of phenomena such as rotating polar phases by soliton spins, SU(2)-DHN breathers, Majorana triplet states, $s\ensuremath{-}p$ mixed dynamics, and so on. These solutions are illustrated by animations, where order parameters are visualized by spherical harmonic functions. The full formulation of the BdG theory is also supported, and the double-counting problem of BdG eigenstates and
N-flavor generalization are discussed.
We consider the linear stability of chiral matter-wave solitons described by a density-dependent gauge theory. By studying the associated Bogoliubov--de Gennes equations both numerically and analytically, we find that the stability problem effectively reduces to that of the standard Gross-Pitaevskii equation, proving that the solitons are stable to linear perturbations. In addition, we formulate the stability problem in the framework of the Vakhitov-Kolokolov criterion and provide supplementary numerical simulations which illustrate the absence of instabilities when the soliton is initially perturbed. These results justify the production of chiral solitons in ultracold experiments and their potential application for practical transport dynamics in interferometry and atomtronics.