Related papers: Symmetries in Connection Preserving Deformations

We wish to explore a link between the Lax integrability of the $q$-Painlev\'e equations and the symmetries of the $q$-Painlev\'e equations. We shall demonstrate that the connection preserving deformations that give rise to the…

We extend Painlev\'e IV model by adding quadratic terms to its Hamiltonian obtaining two classes of models (coalescence and deformation) that interpolate between Painlev\'e IV and II equations for special limits of the underlying…

We present a method of determining a Lax representation for similarity reductions of autonomous and non-autonomous partial difference equations. This method may be used to obtain Lax representations that are general enough to provide the…

We consider the associated linear problem for a q-analogue of the fifth Painleve equation (qPV). We identify a lattice of connection preserving deformations in the space of the connection data for the linear problem with the lattice of…

Folding transformation of the Painlev\'e equations is an algebraic (of degree greater than 1) transformation between solutions of different equations. In 2005 Tsuda, Okamoto and Sakai classified folding transformations of differential…

A $q$-difference analog of the sixth Painlev\'e equation is presented. It arises as the condition for preserving the connection matrix of linear $q$-difference equations, in close analogy with the monodromy preserving deformation of linear…

We study isomonodromic deformation of Fuchsian linear q-difference systems. Furthermore, we are looking of the behaviour of the Birkhoff connexion matrix when q goes to $1$. We use our results to study the convergence of the Birkhoff…

We consider lattice equations on ${\mathds{Z}}^2$ which are autonomous, affine linear and possess the symmetries of the square. Some basic properties of equations of this type are derived, as well as a sufficient linearization condition and…

We introduce the concept of $\omega$-lattice, constructed from $\tau$ functions of Painlev\'e systems, on which quad-equations of ABS type appear. In particular, we consider the $A_5^{(1)}$- and $A_6^{(1)}$-surface $q$-Painlev\'e systems…

Proper lattices for the discrete BKP and the discrete DKP equaitons are determined. Linear B\"acklund transformation equations for the discrete BKP and the DKP equations are constructed, which possesses the lattice symmetries and generate…

The exchange graph of a 2-acyclic quiver is the graph of mutation-equivalent quivers whose edges correspond to mutations. When the quiver admits a nondegenerate Jacobi-finite potential, the exchange graph admits a natural acyclic…

It is well known that the sixth Painlev\'e equation $\PVI$ admits a group of B\"acklund transformations which is isomorphic to the affine Weyl group of type $\mathrm{D}_4^{(1)}$. Although various aspects of this unexpectedly large symmetry…

We examine quantum extensions of the continuous Painlev\'e equations, expressed as systems of first-order differential equations for non-commuting objects. We focus on the Painlev\'e equations II, IV and V. From their auto-B\"acklund…

Backlund transformations are used to search for solutions, particularly soliton solutions, of non-linear differential equations. In this paper we present an invariant geometrical theory of Backlund transformations for second order evolution…

We give new Backlund transformations (BTs) for some known integrable (in the sense of being multidimensionally consistent) quadrilateral lattice equations. As opposed to the natural auto-BT inherent in every such equation, these BTs are of…

We introduce certain B\"acklund transformations for rational solutions of the Painlev\'e VI equation. These transformations act ona family of Painlev\'e VI tau functions. They are obtained from reducing the Hirota bilinear equations that…

The inherent self-consistency properties of the six-point multi-ratio equation allow it to be considered on a domain associated with a T-shaped Coxeter-Dynkin diagram. This extends the KP lattice, which has A_N symmetry, and incorporates…

A geometric study of two 4-dimensional mappings is given. By the resolution of indeterminacy they are lifted to pseudo-automorphisms of rational varieties obtained from $({\mathbb P}^1)^4$ by blowing-up along sixteen 2-dimensional…

We give B\"acklund transformations for first and second Painlev\'e hierarchies. These B\"acklund transformations are generalization of known B\"acklund transformations of the first and second Painlev\'e equations and they relate the…

We identify the self-similarity limit of the second flow of $sl(N)$ mKdV hierarchy with the periodic dressing chain thus establishing % a connection to $A^{(1)}_{N-1}$ invariant Painlev\'e equations. The $A^{(1)}_{N-1}$ B\"acklund…