Binary nonlinearization for the Dirac systems

TL;DR

This paper introduces a binary nonlinearization approach for Dirac systems, transforming their Lax pairs into finite-dimensional integrable Hamiltonian systems, thus revealing their integrability structure and solution representations.

Contribution

It proposes a novel binary nonlinearization method for Dirac systems, linking spectral problems to finite-dimensional integrable Hamiltonian systems.

Findings

Finite-dimensional Liouville integrable Hamiltonian systems derived from Dirac systems.

Hierarchy of commutative Hamiltonian systems with conserved integrals.

Solutions exhibit integrability by quadratures.

Abstract

A Bargmann symmetry constraint is proposed for the Lax pairs and the adjoint Lax pairs of the Dirac systems. It is shown that the spatial part of the nonlinearized Lax pairs and adjoint Lax pairs is a finite dimensional Liouville integrable Hamiltonian system and that under the control of the spatial part, the time parts of the nonlinearized Lax pairs and adjoint Lax pairs are interpreted as a hierarchy of commutative, finite dimensional Liouville integrable Hamiltonian systems whose Hamiltonian functions consist of a series of integrals of motion for the spatial part. Moreover an involutive representation of solutions of the Dirac systems exhibits their integrability by quadratures. This kind of symmetry constraint procedure involving the spectral problem and the adjoint spectral problem is referred to as a binary nonlinearization technique like a binary Darboux transformation.