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By applying Darboux-Crum transformations to the quantum one-gap Lame system,we introduce an arbitrary countable number of bound states into forbiddenbands. The perturbed potentials are reflectionless and contain two types ofsoliton defects in the periodic background. The bound states with finite numberof nodes are supported in the lower forbidden band by the periodicity defectsof the potential well type, while the pulse type bound states in the gap haveinfinite number of nodes and are trapped by defects of the compressionmodulations nature. We investigate the exotic nonlinear N=4 supersymmetricstructure in such paired Schrodinger systems, which extends an ordinary N=2supersymmetry and involves two bosonic generators composed from Lax-Novikovintegrals of the subsystems. One of the bosonic integrals has a nature of acentral charge, and allows us to liaise the obtained systems with thestationary equations of the Korteweg-de Vries and modified Korteweg-de Vrieshierarchies. This exotic supersymmetry opens the way for the construction ofself-consistent condensates based on the Bogoliubov-de Gennes equations andassociated with them new solutions to the Gross-Neveu model. They correspond tothe kink or kink-antikink defects of the crystalline background in dependenceon whether the exotic supersymmetry is unbroken or spontaneously broken.

Soliton defects in one-gap periodic system and exotic supersymmetry