  Reflectionless potentials play an important role in constructing exact
solutions to classical dynamical systems, non-perturbative solutions of various
large-N field theories, and closely related solitonic solutions to the
Bogoliubov-de Gennes equations in the theory of superconductivity. These
solutions rely on the inverse scattering method, which reduces these seemingly
unrelated problems to identifying reflectionless potentials of an auxiliary
one-dimensional quantum scattering problem. There are several ways of
constructing these potentials, one of which is quantum mechanical supersymmetry
(SUSY). In this paper, motivated by recent experimental platforms, we
generalize this framework to develop a theory of solitons in lattice versions
of interacting field theories. Our analysis hinges on the use of trace
identities, which are relations connecting the potential of an equation of
motion to the scattering data. For a discrete Schr\"odinger operator, such
trace identities are well known, and we derive a new set of identities for the
discrete Dirac operator. We then use these in a lattice Gross-Neveu and chiral
Gross-Neveu (Nambu-Jona-Lasinio) model to show that lattice solitons correspond
to reflectionless potentials associated with the discrete scattering problem.
These models are of significance as they are equivalent to a mean-field theory
of a lattice superconductor. To explicitly construct these solitons, we
generalize supersymmetric quantum mechanics to tight-binding models. We show
that a matrix transformation exists that maps a tight-binding model to an
isospectral one which shares the same structure and scattering properties. The
corresponding soliton solutions have both modulated hopping and onsite
potential; we explicitly compute both topological and non-topological soliton
solutions as well as bound state spectra in the aforementioned models.
