[Paper Review] Binary nonlinearization for the Dirac systems

Wen‐Xiu Ma|ArXiv.org|Dec 6, 1995

Spectral Theory in Mathematical Physics3 references21 citations

TL;DR

This paper introduces a binary nonlinearization technique for Dirac systems by imposing a Bargmann symmetry constraint on both eigenfunctions and adjoint eigenfunctions, transforming the Lax pairs into finite-dimensional Liouville integrable Hamiltonian systems. The key result is an involutive representation of solutions that exhibits integrability by quadratures, demonstrating the system's complete integrability through Hamiltonian flows and conserved quantities.

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.

Motivation & Objective

• To develop a novel symmetry constraint method—binary nonlinearization—for Dirac-type integrable systems.
• To establish a connection between the Lax pairs and adjoint Lax pairs of the Dirac system and finite-dimensional Hamiltonian systems.
• To demonstrate that the constrained systems are Liouville integrable and admit solutions expressible by quadratures.
• To generalize the nonlinearization technique by incorporating both eigenfunctions and adjoint eigenfunctions, analogous to binary Darboux transformations.

Proposed method

• A Bargmann symmetry constraint is imposed on the spectral problem and its adjoint, linking the eigenfunctions and adjoint eigenfunctions to the spectral parameter's variational derivative.
• The spatial part of the nonlinearized Lax pairs is shown to form a finite-dimensional Liouville integrable Hamiltonian system with Hamiltonian function H.
• The time parts of the nonlinearized Lax pairs are interpreted as commutative, finite-dimensional Liouville integrable systems with Hamiltonian functions F_n derived from integrals of motion of the spatial part.
• The involutive representation of solutions is constructed using Hamiltonian phase flows g^x_H and g^{t_n}_{F_{n+1}} associated with H and F_{n+1}, respectively.
• The constraint c² + a² - b² = 1 is derived from the trace of V², ensuring the consistency of the nonlinearization.
• The method extends the standard nonlinearization by including both the original and adjoint spectral problems, forming a binary structure.

Experimental results

Research questions

• RQ1 How can a symmetry constraint be formulated that simultaneously involves both eigenfunctions and adjoint eigenfunctions in the Dirac system?
• RQ2 Can the nonlinearized Lax pairs and adjoint Lax pairs be shown to form finite-dimensional Liouville integrable Hamiltonian systems?
• RQ3 What is the role of the Hamiltonian functions F_n in the time evolution of the constrained system?
• RQ4 How does the involutive representation of solutions demonstrate integrability by quadratures?
• RQ5 What is the relationship between the binary nonlinearization and the standard nonlinearization or binary Darboux transformation?

Key findings

• The spatial part of the nonlinearized Lax pairs and adjoint Lax pairs forms a finite-dimensional Liouville integrable Hamiltonian system with Hamiltonian function H.
• The time parts of the nonlinearized Lax pairs are commutative, finite-dimensional Liouville integrable systems whose Hamiltonian functions F_n consist of integrals of motion from the spatial part.
• An involutive representation of solutions is derived, showing that the Dirac system's solutions can be expressed through Hamiltonian phase flows g^x_H and g^{t_n}_{F_{n+1}}.
• The constraint c² + a² - b² = 1 holds identically, ensuring the consistency of the nonlinearization and the closure of the system.
• The binary nonlinearization technique generates a large class of finite-dimensional Liouville integrable Hamiltonian systems, extending the standard nonlinearization method.
• Under specific reductions, such as Ψ₁ = -Φ₂, Ψ₂ = Φ₁, the system reduces to a form compatible with the standard nonlinearization, confirming the broader applicability of the binary approach.