This laboratory focuses on soliton equations and Painlev\'e equations. As is well known, the soliton equations are reduced to the Painlev\'e equations by suitable reduction procedures. By the so-called similarity reduction, all the Painlev\'e equations except for type-I can be derived from the soliton equations.

By the way, discrete Painlev\'e equations can be understood uniformly in the framework of algebraic geometry based on the rational varieties. It is expected that these discrete Painlev\'e equations can also be derived from the discrete soliton equations by the similarity reduction.

My interest is to examine the solution structures and the affine Weyl group symmetries of the discrete Painlev\'e equations in terms of the discrete soliton equations. I am working on the similarity reduction of the soliton equations associated to the toroidal Lie algebras and a construction of the discrete soliton equations corresponding to the discrete Painlev\'e equations which have the affine Weyl group symmetries of type-E.