Twisted kinks, Dirac transparent systems and Darboux transformations

TL;DR

This paper uses Darboux transformations to construct and analyze reflectionless Dirac systems with pseudoscalar potentials, revealing their solitonic structure and spectral properties, and connecting them to well-known models and integrable hierarchies.

Contribution

It introduces a four-parameter class of reflectionless Dirac systems with twisted kink potentials, linking them to integrable hierarchies and transparent non-relativistic models.

Findings

Constructed a four-parameter family of reflectionless Dirac systems.

Linked twisted kinks to multi-solitonic solutions of the AKNS hierarchy.

Identified spectral properties and conserved quantities of the systems.

Abstract

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 Schr\"odinger Hamiltonian is studied and the rich algebraic structure of their integrals of motion is discussed.