相关论文: Affine Weyl group approach to Painlev\'e equations

We derive integrable equations starting from autonomous mappings with a general form inspired by the multiplicative systems associated to the affine Weyl group E$_8^{(1)}$. Five such systems are obtained, three of which turn out to be…

We propose a method to characterize discrete time evolution equations, which generalize discrete time soliton equations, including the $q$-difference Painlev\'e IV equations discussed recently by Kajiwara, Noumi and Yamada.

We present an new system of ordinary differential equations with affine Weyl group symmetry of type E_6^{(1)}. This system is expressed as a Hamiltonian system of sixth order with a coupled Painleve VI Hamiltonian.

Higher order Painleve equations invariant under extended affine Weyl groups $A^{(1)}_n$ are obtained through self-similarity limit of a class of pseudo-differential Lax hierarchies with symmetry inherited from the underlying generalized…

Recently, a birational representation of an extended affine Weyl group of $(A_{2N}\rtimes A_1)^{(1)}$-type, which gives a higher-order generalization of an $A_4^{(1)}$-surface type $q$-Painlev\'e equation, was obtained. In this paper, we…

This survey article, is written as an extended note and supplement of my lectures in the current developments in mathematics conference in 2015. We discuss some recent developments on the conjugacy classes of affine Weyl groups and $p$-adic…

We define certain extensions of affine Weyl groups (distinct from these considered by K. Saito [S1] in the theory of extended affine root systems), prove an analogue of Chevalley theorem for their invariants, and construct a Frobenius…

In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlev\'e equations, with a particular emphasis on the discrete Painlev\'e equations. The theory is controlled by the…

We construct a tau cover of the generalized Drinfeld-Sokolov hierarchy associated to an arbitrary affine Kac-Moody algebra with gradations $\mathrm{s}\le\mathds{1}$ and derive its Virasoro symmetries. By imposing the Virasoro constraints we…

We investigate some of the discrete Painleve equations (dPII, qPI and qPII) and the discrete KdV equation over finite fields. The first part concerns the discrete Painleve equations. We review some of the ideas introduced in our previous…

We show that the Garnier system in n-variables has affine Weyl group symmetry of type $B^{(1)}_{n+3}$. We also formulate the $\tau$ functions for the Garnier system (or the Schlesinger system of rank 2) on the root lattice $Q(C_{n+3})$ and…

Recently a certain $q$-Painlev\'e type system has been obtained from a reduction of the $q$-Garnier system. In this paper it is shown that the $q$-Painlev\'e type system is associated with another realization of the affine Weyl group…

The generalized Drinfeld-Sokolov construction of KdV systems is reviewed in the case of an arbitrary affine Lie algebra paying particular attention to Hamiltonian aspects and $\W$-algebras. Some extensions of known results as well as a new…

A discretization of Painlev\'e VI equation was obtained by Jimbo and Sakai in 1996. There are two ways to quantize it: 1) use the affine Weyl group symmetry (of $D_5^{(1)}$) (Hasegawa, 2011), 2) Lax formalism i.e. monodromy preserving point…

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…

Since the classification of discrete Painlev\'e equations in terms of rational surfaces, there has been much interest in the range of integrable equations arising from each of the 22 surface types in Sakai's list. For all but the most…

In this paper, we find a two-parameter family of coupled Painlev\'e systems in dimension four with affine Weyl group symmetry of type $D_3^{(2)}$. We also find a four-parameter family of 2-coupled $D_3^{(2)}$-systems in dimension eight with…

In this paper, we show how to relate $n$-dimensional cubes on which ABS equations hold to the symmetry groups of discrete Painlev\'e equations. We here focus on the reduction from the 4-dimensional cube to the $q$-discrete third Painlev\'e…

We construct certain Steinberg groups associated to extended affine Lie algebras and their root systems. Then by the integration methods of Kac and Peterson for integrable Lie algebras, we associate a group to every tame extended affine Lie…

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…