On the Lagrangian Structure of the Discrete Isospectral and Isomonodromic Transformations

We establish the Lagrangian nature of the discrete isospectral and isomonodromic dynamical systems corresponding to the re-factorization transformations of the rational matrix functions on the Riemann sphere. Specifically, in the isospectral case we generalize the Moser-Veselov approach to integrability of discrete systems via the re-factorization of matrix polynomials to a more general class of matrix rational functions that have a simple divisor and, in the quadratic case, explicitly write the Lagrangian function for such systems. Next we show that if we let certain parameters in this Lagrangian to be time-dependent, the resulting Euler-Lagrange equations describe the isomonodromic transformations of systems of linear difference equations. It is known that in some special cases such equations reduce to the difference Painlev'e equation. As an example, we show how to obtain the difference Painlev`e V equation in this way, and hence we establish that this equation can be written in the Lagrangian form.

💡 Research Summary

The paper investigates discrete dynamical systems that arise from the re‑factorization of rational matrix functions on the Riemann sphere and shows that both the isospectral and the isomonodromic versions of these systems possess a genuine Lagrangian formulation. Starting from a rational matrix‑valued function (L(z)=A(z)B(z)^{-1}) with a simple divisor, the authors factor it as (L(z)=L_{-}(z)L_{+}(z)) and consider the elementary transformation that swaps the order of the factors: \