By applying Darboux-Crum transformations to the quantum one-gap Lame system,
we introduce an arbitrary countable number of bound states into forbidden
bands. The perturbed potentials are reflectionless and contain two types of
soliton defects in the periodic background. The bound states with finite number
of nodes are supported in the lower forbidden band by the periodicity defects
of the potential well type, while the pulse type bound states in the gap have
infinite number of nodes and are trapped by defects of the compression
modulations nature. We investigate the exotic nonlinear N=4 supersymmetric
structure in such paired Schrodinger systems, which extends an ordinary N=2
supersymmetry and involves two bosonic generators composed from Lax-Novikov
integrals of the subsystems. One of the bosonic integrals has a nature of a
central charge, and allows us to liaise the obtained systems with the
stationary equations of the Korteweg-de Vries and modified Korteweg-de Vries
hierarchies. This exotic supersymmetry opens the way for the construction of
self-consistent condensates based on the Bogoliubov-de Gennes equations and
associated with them new solutions to the Gross-Neveu model. They correspond to
the kink or kink-antikink defects of the crystalline background in dependence
on whether the exotic supersymmetry is unbroken or spontaneously broken.Comment: 44 pages, 11 figures; comments and refs added, version to appear in
Phys. Rev.
