The Lax Integrable Differential-Difference Dynamical Systems on Extended Phase Spaces

The Hamiltonian representation for the hierarchy of Lax-type flows on a dual space to the Lie algebra of shift operators coupled with suitable eigenfunctions and adjoint eigenfunctions evolutions of associated spectral problems is found by means of a specially constructed Backlund transformation. The Hamiltonian description for the corresponding set of squared eigenfunction symmetry hierarchies is represented. The relation of these hierarchies with Lax integrable (2+1)-dimensional differential-difference systems and their triple Lax-type linearizations is analysed. The existence problem of a Hamiltonian representation for the coupled Lax-type hierarchy on a dual space to the central extension of the shift operator Lie algebra is solved also.

💡 Research Summary

The paper addresses the long‑standing problem of constructing a Hamiltonian description for Lax‑type integrable differential‑difference dynamical systems when the phase space is enlarged by the inclusion of eigenfunctions and adjoint eigenfunctions associated with the underlying spectral problem. The authors begin by recalling the Lie algebra 𝔤 of shift (or translation) operators on a one‑dimensional lattice and its dual space 𝔤*. On 𝔤* a natural Lie–Poisson (Kirillov–Kostant) structure exists, and the standard Lax equation
  L̇ =