**Author:**
Saburo Kakei,
Tetsuya Kikuchi

**Comments:** English, English summary

**Journal-ref:**
Preprint

A q-alalogue ofgl_{3}Drinfel'd-Sokolov hierarchy is proposed. Applying similarity reduction to the hierarchy, one can obtain theq-Painleve VI equation, proposed by Jimbo and Sakai.

**Author:**
Saburo Kakei,
Tetsuya Kikuchi

**Comments:** English, English summary

**Journal-ref:**
Preprint

Scaling symmetry ofgl_{n}-type Drinfel'd-Sokolov hierarchy is investigated. Applying similarity reduction to the hierarchy, one can obtain the Schlesinger equation with (n+1) regular singularities. Especially in the case ofn=3, the hierarchy contains the three-wave resonant system and the similarity reduction gives the generic case of the Painleve VI equation. We also discuss Weyl group symmetry of the hierarchy.

**Author:**
Saburo Kakei, Tetsuya Kikuchi

**Comments:** English, English summary

**Journal-ref:**
Glasgow Mathematical Journal, 47A (2005), 99-107

Hierarchy structure of a derivative nonlinear Schrödinger equation is investigated in terms of Sato-Segal-Wilson formulation. Special solutions are constructed as ratio of Wronski determinants. Relations to the Painlevé IV and the discrete Painlevé I are discussed by applying a similarity reduction.

**Author:**
Saburo Kakei,
Tetsuya Kikuchi

**Comments:** English, English summary

**Journal-ref:**
Int. Math. Res. Not. 78 (2004), 4181-4209

The generalized Drinfel'd-Sokolov hierarchies studied by de Groot-Hollowood-Miramontes are extended by the viewpoint of Sato-Wilson dressing method. In theA_{1}^{(1)}case, we obtain the hierarchy which include the derivative nonlinear Schrödinger equation. We give two types of affine Weyl group symmetry of the hierarchy based on the Gauss decomposition of an affine Lie group. As a similarity reduction, the fourth Painlevé equation and their Weyl group symmetry are obtained. We also clarify the connection between these systems and Lax formulation of Painlevé system based on monodromy preserving deformations.

**Author:**
Saburo Kakei

**Comments:** English, English summary

**Journal-ref:**
Preprint

Bilinear identity associated with the self-dual Yang-Mills hierarchy is discussed by using a fermionic representation of the toroidal Lie algebrasl^{tor}_{2}.

**Author:**
Tetsuya Kikuchi, Takeshi Ikeda, Saburo Kakei

**Comments:** English, English summary

**Journal-ref:**
J. Phys. ** A36** (2003) 11465-11480

We study a similarity reduction of the modified Yajima-Oikawa hierarchy. The hierarchy is associated with a non-standard Heisenberg subalgebra in the affine Lie algebra of typeA_{2}^{(1)}. The system of equations for self-similar solutions is presented as a Hamiltonian system of degree of freedom two, and admits a group of Bäcklund transformations isomorphic to the affine Weyl group of typeA_{2}^{(1)}. We show that the system is equivalent to a two-parameter family of the fifth Painlevé equation.

Self-Dual Yang-Mills Equation, and Toroidal Lie Algebras

**Author:**
Saburo Kakei, Takeshi Ikeda and Kanehisa Takasaki

**Comments:** English, English summary

**Journal-ref:**
Annales Henri Poincaré ** 3** (2002) 817-845

The hierarchy structure associated with a (2+1)-dimensional nonlinear Schrödinger equation is discussed as an extension of the theory of the KP hierarchy. Several methods to construct special solutions are given. The relation between the hierarchy and a representation of toroidal Lie algebras are established by using the language of free fermions. A relation to the self-dual Yang-Mills equation is also discussed.

**Author:**
Saburo Kakei and Yasuhiro Ohta

**Comments:** English, English summary

**Journal-ref:**
J. Phys. ** A34** (2001) 10585-10592

We present a novel differential-difference system in (2+1)-dimensional space-time (one discrete, two continuum), arisen from the Bogoyavlensky's (2+1)-dimensional KdV hierarchy. Our method is based on the bilinear identity of the hierarchy, which is related to the vertex operator representation of the toroidal Lie algebrasl^{tor}_{2}.

**Authors:**
Saburo Kakei and Yusuke Kato

**Comments:** English, English summary

**Journal-ref:**
In *SPECIAL FUNCTIONS -Proceedings of the International Workshop-*, Eds.:
C. Dunkl, M. Ismail and R. Wong, World Scientific, Singapore, 2000, pp. 125-139.

Algebraic treatment of the multivariable orthogonal polynomials associated with the quantum Calogero models are reviewed. Several explicit formulas involving a differential formula of Jack polynomials are deduced from algebraic structure.

**Author:** Saburo
Kakei

**Comments:** English, English summary

**Journal-ref:** Phys.
Lett. **A264** (2000) 449-458

The coupled KP hierarchy, introduced by Hirota and Ohta, are investigated by using the dressing method. It is shown that the coupled KP hierarchy can be reformulated as a reduced case of the 2-component KP hierarchy.

**Author:**
Saburo Kakei

**Comments:** English, English summary

**Journal-ref:**
J. Phys. Soc. Jpn. **68** (1999) 2875-2877

Orthogonal and symplectic matrix integrals are investigated. It is shown that the matrix integrals can be considered as a τ-function of the coupled KP hierarchy, whose solution can be expressed in terms of pfaffians.

**Author:**
Saburo Kakei

**Comments:** English, English summary

**Journal-ref:**
J. Math. Phys. **39** (1998) 4993-5006

Several properties of the multivariable Hermite and Laguerre polynomials associated with the quantum Calogero models are investigated by using the operators that intertwine representations of a degenerate version of the double affine Hecke algebra. As applications, raising operators and shift operators for the polynomials are constructed in unified manner.

**Author:**
Saburo Kakei

**Comments:** English, English summary

**Journal-ref:**
J. Phys. ** A30** (1997) L535-L541

We investigate algebraic structure for theB-type Calogero model by using the exchange-operator formalism. We show that the set of the Jack polynomials whose arguments are Dunkl-type operators provides an orthogonal basis._{N}

**Author:**
Saburo Kakei

**Comments:** English, English summary

**Journal-ref:**
J. Phys. ** A29** (1996) L619-L624

We investigate common algebraic structure for the rational and trigonometric Calogero-Sutherland models by using the exchange-operator formalism. We show that the set of the Jack polynomials whose arguments are Dunkl-type operators provides an orthogonal basis for the rational case.

**Author:**
Saburo Kakei

**Comments:** English, English summary

**Journal-ref:**
J. Phys. Soc. Jpn. **65** (1996) 337-339

In this letter, we show that certain Fredholm determinantD(λ;t), introduced by Zamolodchikov in his study of 2D polymers, is a continuum limit of soliton solution for the Toda lattice hierarchy with 2-periodic reduction condition.

**Authors:**
Saburo Kakei, Narimasa Sasa and Junkichi Satsuma

**Comments:** English,
English summary

**Journal-ref:** J. Phys. Soc. Jpn. **64** (1995) 1519-1523

A generalized derivative nonlinear Schrodinger equation,i

q+_{t}q+ 2i_{xx}γ|q|^{2}q+ 2i (_{x}γ-1)q^{2}q^{*}+ (_{x}γ-1)(γ-2)|q|^{4}q= 0 ,is studied by means of Hirota's bilinear formalism. Soliton solutions are constructed as quotients of Wronski-type determinants. A relationship between the bilinear structure and gauge transformation is also discussed.

**Authors:**
Saburo Kakei and Junkichi Satsuma

**Comments:** English, English summary

**Journal-ref:**
J. Phys. Soc. Jpn. **63** (1994) 885-894

Multi-soliton solutions of a coupled system of the Nonlinear Schrodinger equation and the Maxwell-Bloch equations are given. Solutions of the system are explicitly constructed as a quatient of Casorati-type determinants. By using explicit form of the soliton solutions, the influence of the MB-term on the speed of soliton is evaluated.