  We construct a fermion system in a multi-kink-antikink soliton background,
and present in an explicit form all its trapped configurations (bound state
solutions) as well as scattering states. This is achieved by exploiting an
exotic N=4 centrally extended nonlinear supersymmetry of completely isospectral
pairs of reflectionless Schrodinger systems with potentials to be n-soliton
solutions for the Korteweg-de Vries equation. The obtained reflectionless Dirac
system with a position-dependent mass is shown to possess its own exotic
nonlinear supersymmetry associated with the matrix Lax-Novikov operator being a
Darboux-dressed momentum. In the process, we get an algebraic recursive
representation for the multi-kink-antikink backgrounds, and establish their
relation to the the modified Korteweg-de Vries equation. We also indicate how
the results can be related to the physics of self-consistent condensates based
on the Bogoliubov-de Gennes equations.
