Abstract
By applying Darboux--Crum transformations to the quantum one-gap Lam\'e 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 a 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 an infinite number of nodes and are trapped by defects of the compression modulations nature. We investigate the exotic nonlinear $\mathcal{N}=4$ supersymmetric structure in such paired Schr\"odinger systems, which extends an ordinary $\mathcal{N}=2$ 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.
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