Transmutations of Supersymmetry Through Soliton Scattering and Self-Consistent Condensates

We consider the two most general families of the (1+1)D Dirac systems with transparent scalar potentials and two related families of the paired reflectionless Schrödinger operators. The ordinary N=2 supersymmetry for such Schrödinger pairs is enlarged up to an exotic N=4 nonlinear centrally extended supersymmetric structure, which involves two bosonic integrals composed from the Lax-Novikov operators for the stationary Korteweg-de Vries hierarchy. Each associated single Dirac system displays a proper N=2 nonlinear supersymmetry with a nonstandard grading operator. One of the two families of the first- and second-order systems exhibits the unbroken supersymmetry, while another is described by the broken exotic supersymmetry. The two families are shown to be mutually transmuted by applying a certain limit procedure to the soliton scattering data. We relate the topologically trivial and nontrivial transparent potentials with self-consistent inhomogeneous condensates in the Bogoliubov-de Gennes and Gross-Neveu models and indicate the exotic N=4 nonlinear supersymmetry of the paired reflectionless Dirac systems. © 2014 American Physical Society.