Integrable discrete systems on R and related dispersionless systems

The general framework for integrable discrete systems on R in particular containing lattice soliton systems and their q-deformed analogues is presented. The concept of regular grain structures on R, generated by discrete one-parameter groups of diffeomorphisms, through which one can define algebras of shift operators is introduced. Two integrable hierarchies of discrete chains together with bi-Hamiltonian structures are constructed. Their continuous limit and the inverse problem based on the deformation quantization scheme are considered.

💡 Research Summary

The paper develops a comprehensive framework for integrable discrete systems defined on the real line ℝ, extending beyond traditional uniform lattices to encompass a broad class of non‑uniform, deformation‑compatible discretizations. The central construct is a “regular grain structure,” generated by a one‑parameter group of smooth diffeomorphisms σt:ℝ→ℝ (t∈ℝ). Each σt is invertible and depends continuously on the parameter, thereby providing a generalized notion of a lattice cell whose spacing Δx(x)=σ1(x)−x may vary with position. This structure allows the definition of shift operators Eσ and difference operators Dσ=Eσ−1, which together generate a non‑commutative algebra Aσ of formal series in powers of Eσ with smooth coefficient functions.

Two compatible Poisson brackets Π0 and Π1 are introduced on Aσ. Π0 is the canonical bracket inherited from the underlying difference algebra, while Π1 incorporates the differential structure of σt, yielding a nonlinear Poisson structure. Their compatibility endows Aσ with a bi‑Hamiltonian hierarchy. Using these ingredients, the authors construct two integrable chains:

1. A σ‑difference KP hierarchy, with Lax operator
   L = Eσ + u0 + u1Eσ⁻¹ + u2Eσ⁻² + …,
   evolving according to ∂_{t_n}L =