On some new forms of lattice integrable equations

Inspired by the forms of delay-Painleve equations, we consider some new differential-discrete systems of KdV, mKdV and Sine-Gordon - type related by simple one way Miura transformations to classical ones. Using Hirota bilinear formalism we construct their new integrable discretizations, some of them having higher order. In particular, by this procedure, we show that the integrable discretization of intermediate sine-Gordon equation is exactly lattice mKdV and also we find a bilinear form of the recently proposed lattice Tzitzeica equation. Also the travelling wave reduction of these new lattice equations is studied and it is shown that all of them, including the higher order ones, can be integrated to Quispel-Roberts-Thomson (QRT) mappings.

💡 Research Summary

The paper introduces a family of new differential‑discrete integrable equations inspired by the structure of delay‑Painlevé equations and shows how they can be systematically discretized into lattice equations using Hirota’s bilinear formalism. The authors start from three semidiscrete models—KdV‑type, mKdV‑type and an intermediate sine‑Gordon‑type—each of which is related to a classical integrable system by a simple one‑way Miura transformation.

For the KdV‑type case they propose the semidiscrete equation
(\dot w_n = w_n w_{n-1}(w_{n+1}-w_{n-1})).
Through the Miura map (u_n = w_n/(w_n-1)) this reduces to the well‑known semidiscrete KdV equation (\dot u_n - u_n^2(u_{n+1}-u_{n-1})=0). By setting (w_n = F_{n+1}/F_n) the Hirota bilinear form (D_tF_{n+1}\cdot F_n = F_{n+2}F_{n-1}-F_nF_{n+1}) is obtained, which is exactly the bilinear representation of the classical semidiscrete KdV. Discretising the time derivative with a step (\delta) and imposing gauge invariance yields a quadrilateral lattice equation (4). Although this lattice equation is not 3‑dimensional consistent (hence not in the ABS list), a Miura transformation (u_{m,n}=W_{m+1,n}/(W_{m,n}-1)) maps it to Hirota’s lattice KdV, establishing the integrable nature of the new lattice model.

The mKdV‑type construction follows an analogous route. The authors introduce
(\dot v_n = 2v_n v_{n+1} - v_{n-1}v_{n+1} + v_{n-1}).
With the Miura map (u_n = i,2,\partial_t\log v_n) this becomes the standard semidiscrete mKdV (\dot u_n = (1+u_n^2)(u_{n+1}-u_{n-1})). Using (v_n = G_n/F_n) leads to the bilinear pair
(D_t G_n\cdot F_n = G_{n+1}F_{n-1} - G_{n-1}F_{n+1}),
(2G_nF_n = G_{n+1}F_{n-1}+G_{n-1}F_{n+1}).
The authors explicitly construct a three‑soliton solution, confirming the integrability of the bilinear system. After discretising time, they obtain a fully discrete bilinear system (16)–(17) and, after eliminating auxiliary variables, derive a higher‑order nonlinear lattice equation (22). This equation is a new, fourth‑order lattice mKdV that still admits a three‑soliton solution, demonstrating that the Hirota method can generate higher‑order integrable lattice equations.

The intermediate sine‑Gordon equation is treated next. Starting from the differential‑difference relation
(\frac{d}{dt}(u_n u_{n+1}) = \gamma u_n^2 + \kappa u_{n+1}^2)
and choosing (\gamma=-\kappa=1) the authors recover the intermediate sine‑Gordon equation, which can be written in terms of a singular integral operator (T). By the substitution (u_n = G_n/F_n) they obtain a bilinear system (24)–(25). Discretising and setting the gauge constant (A=1) leads to bilinear equations (26)–(27). Eliminating the ratio (X_{m,n}=G_{m,n}/F_{m,n}) yields the classical lattice mKdV equation, showing that the lattice version of the intermediate sine‑Gordon is exactly the lattice mKdV (up to a simple exponential re‑normalisation). This reinforces the well‑known equivalence between lattice mKdV and lattice sine‑Gordon.

A further contribution is the bilinear formulation of the recently proposed lattice Tzitzeica equation. Starting from a general differential‑difference equation (30) that encompasses various bi‑Riccati delay‑Painlevé III reductions, the authors introduce (u=G/F) and derive bilinear equations (31)–(32). After a careful choice of parameters and discretisation, they obtain the pair (35)–(36). Defining (W_{m,n}=G_{m,n}/F_{m,n}) these equations reduce precisely to Adler’s lattice Tzitzeica equation. Moreover, by expressing (G) and (F) through tau‑functions, the authors show that (35)–(36) correspond to the trilinear form found by Adler, confirming that they have identified the Hirota bilinear representation of the lattice Tzitzeica system.

The final part of the paper analyses travelling‑wave reductions of all the lattice equations. By imposing the reduction (W_{m,n}=x_{n+m}) (or analogous reductions for the other variables) each quadrilateral lattice equation collapses to a one‑dimensional recurrence. For the KdV‑type lattice (4) the reduction yields a mapping that can be rewritten as a symmetric QRT map, an additive type associated with the autonomous limit of a (q)-Painlevé equation. The lattice mKdV (6) gives a multiplicative QRT map, while the higher‑order lattice mKdV (22) leads to a fourth‑order QRT mapping. All these mappings possess a bi‑quadratic invariant and are integrable via elliptic functions, confirming that the newly constructed lattice equations belong to the QRT class after reduction.

In summary, the authors have built a coherent framework that starts from delay‑Painlevé‑inspired semidiscrete equations, uses Hirota’s bilinear method to generate both known and novel lattice integrable systems (including higher‑order ones), establishes Miura connections among them, provides explicit three‑soliton solutions as integrability certificates, and demonstrates that their travelling‑wave reductions are precisely QRT mappings. This work enriches the catalogue of integrable lattice equations, offers new higher‑order examples, and clarifies the algebraic structure linking delay‑Painlevé equations, Hirota bilinear forms, and QRT dynamics.