Discrete Painlevé equation, Miwa variables and string equation in 5d matrix models

Unitary One Matrix Models: String Equation and Flows

We review the Symmetric Unitary One Matrix Models. In particular we discuss the string equation in the operator formalism, the mKdV flows and the Virasoro Constraints. We focus on the τ -function formalism for the flows and we describe its connection to the (big cell of the) Sato Grassmannian Gr via the Plucker embedding of Gr into a fermionic Fock space. Then the space of solutions to the stri...

Desargues Maps and the Hirota–miwa Equation

We study the Desargues maps φ : Z → P , which generate lattices whose points are collinear with all their nearest (in positive directions) neighbours. The multidimensional consistency of the map is equivalent to the Desargues theorem and its higher-dimensional generalizations. The nonlinear counterpart of the map is the non-commutative (in general) Hirota–Miwa system. In the commutative case of...

On Discrete Painlevé Equations Associated with the Lattice Kdv Systems and the Painlevé Vi Equation

1 Abstract A new integrable nonautonomous nonlinear ordinary difference equation is presented which can be considered to be a discrete analogue of the Painlevé V equation. Its derivation is based on the similarity reduction on the two-dimensional lattice of integrable partial difference equations of KdV type. The new equation which is referred to as GDP (generalised discrete Painlevé equation) ...

Errors in Variables in Simultaneous Equation Models

On Reductions of the Hirota–Miwa Equation

The Hirota–Miwa equation (also known as the discrete KP equation, or the octahedron recurrence) is a bilinear partial difference equation in three independent variables. It is integrable in the sense that it arises as the compatibility condition of a linear system (Lax pair). The Hirota–Miwa equation has infinitely many reductions of plane wave type (including a quadratic exponential gauge tran...