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Twisted kinks, Dirac transparent systems and Darboux transformations

Darboux transformations are employed in construction and analysis of Dirac Hamiltonians with pseudoscalar potentials. By this method, we build a four parameter class of reflectionless systems. Their potentials correspond to composition of complex kinks, also known as twisted kinks, that play an important role in the $1+1$ Gross-Neveu and Nambu-Jona-Lasinio field theories. The twisted kinks turn out to be multi-solitonic solutions of the integrable AKNS hierarchy. Consequently, all the spectral properties of the Dirac reflectionless systems are reflected in a non-trivial conserved quantity, which can be expressed in a simple way in terms of Darboux transformations. We show that the four parameter pseudoscalar systems reduce to well-known models for specific choices of the parameters. An associated class of transparent non-relativistic models described by matrix Schrödinger Hamiltonian is studied and the rich algebraic structure of their integrals of motion is discussed.