Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFactorizations of Rational Matrix Functions with Application to Discrete Isomonodromic Transformations and Difference Painleve Equations
165
0
0.0
(
0
)
Authors
Anton Dzhamay
Ask ChatGPT about the research
No Arabic abstract
We study factorizations of rational matrix functions with simple poles on the Riemann sphere. For the quadratic case (two poles) we show, using multiplicative representations of such matrix functions, that a good coordinate system on this space is given by a mix of residue eigenvectors of the matrix and its inverse. Our approach is motivated by the theory of discrete isomonodromic transformations and their relationship with difference Painleve equations. In particular, in these coordinates, basic isomonodromic transformations take the form of the discrete Euler-Lagrange equations. Secondly we show that dPV equations, previously obtained in this context by D. Arinkin and A. Borodin, can be understood as simple relationships between the residues of such matrices and their inverses.
rate research
Read More
We express discrete Painleve equations as discrete Hamiltonian systems. The discrete Hamiltonian systems here mean the canonical transformations defined by generating functions. Our construction relies on the classification of the discrete Painleve equations based on the surface-type. The discrete Hamiltonians we obtain are written in the logarithm and dilogarithm functions.
Schlesinger transformations are algebraic transformations of a Fuchsian system that preserve its monodromy representation and act on the characteristic indices of the system by integral shifts. One of the important reasons to study such transformations is the relationship between Schlesinger transformations and discrete Painleve equations; this is also the main theme behind our work. We derive emph{discrete Schlesinger evolution equations} describing discrete dynamical systems generated by elementary Schlesinger transformations and give their discrete Hamiltonian description w.r.t.~the standard symplectic structure on the space of Fuchsian systems. As an application, we compute explicitly two examples of reduction from Schlesinger transformations to difference Painleve equations. The first example, d-$Pbig(D_{4}^{(1)}big)$ (or difference Painleve V), corresponds to Backlund transformations for continuous $P_{text{VI}}$. The second example, d-$Pbig(A_{2}^{(1)*}big)$ (with the symmetry group $E_{6}^{(1)}$), is purely discrete. We also describe the role played by the geometry of the Okamoto space of initial conditions in comparing different equations of the same type.
We present two examples of reductions from the evolution equations describing discrete Schlesinger transformations of Fuchsian systems to difference Painleve equations: difference Painleve equation d-$Pleft({A}_{2}^{(1)*}right)$ with the symmetry group ${E}^{(1)}_{6}$ and difference Painleve equation d-$Pleft({A}_{1}^{(1)*}right)$ with the symmetry group ${E}^{(1)}_{7}$. In both cases we describe in detail how to compute their Okamoto space of the initial conditions and emphasize the role played by geometry in helping us to understand the structure of the reduction, a choice of a good coordinate system describing the equation, and how to compare it with other instances of equations of the same type.
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 Painleve 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.
Iorgov, Lisovyy, and Teschner established a connection between isomonodromic deformation of linear differential equations and Liouville conformal field theory at $c=1$. In this paper we present a $q$ analog of their construction. We show that the general solution of the $q$-Painleve VI equation is a ratio of four tau functions, each of which is given by a combinatorial series arising in the AGT correspondence. We also propose conjectural bilinear equations for the tau functions.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
