A q-analogue of gl₃ hierarchy and q-Painleve VI

Author: Saburo Kakei, Tetsuya Kikuchi
Comments: English, English summary
Journal-ref: Preprint

A q-alalogue of gl₃ Drinfel'd-Sokolov hierarchy is proposed. Applying similarity reduction to the hierarchy, one can obtain the q-Painleve VI equation, proposed by Jimbo and Sakai.

The sixth Painleve equation as similarity reduction of gl₃ hierarchy

Author: Saburo Kakei, Tetsuya Kikuchi
Comments: English, English summary
Journal-ref: Preprint

Scaling symmetry of gl_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 of n=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.

Toroidal Lie Algebra and Bilinear Identity of the Self-Dual Yang-Mills Hierarchy

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 algebra sl^tor₂.

Solutions of a Derivative Nonlinear Schrodinger Hierarchy and Its Similarity Reduction

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.

Affine Lie Group Approach to a Derivative Nonlinear Schrodinger Equation and Its 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 the A₁⁽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.

Similarity Reduction of the Modified Yajima-Oikawa Equation

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 type A₂⁽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 type A₂⁽¹⁾. We show that the system is equivalent to a two-parameter family of the fifth Painlevé equation.

Hierarchy of (2+1)-dimensional nonlinear Schrödinger equation, 
self-dual Yang-Mills equation, and toroidal Lie algebras

Author: Saburo Kakei, Takeshi Ikeda and Kanehisa Takasaki
Comments: English, English summary
Journal-ref: preprint (nlin.SI/0107065)

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. 

Differential-difference system related to toroidal Lie algebra

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 algebra sl^tor₂.

Algebraic Aspects of Quantum Calogero Models

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.

Dressing Method and the Coupled KP Hierarchy

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.

Orthogonal and Symplectic Matrix Integrals and Coupled 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. 

Intertwining Operators for a Degenerate Double Affine Hecke Algebra and Multivariable Orthogonal Polynomials

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.

An Orthogonal Basis for the B_N-type Calogero Model

Author: Saburo Kakei
Comments: English, English summary
Journal-ref: J. Phys. A30 (1997) L535-L541

We investigate algebraic structure for the B_N-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.

Common Algebraic Structure for the Calogero-Sutherland Models

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.

Toda Lattice Hierarchy and Zamolodchikov's Conjecture

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 determinant D(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.

Bilinearization of a Generalized Derivative Nonlinear Schrödinger Equation

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_xx + 2i γ|q|²q_x + 2i (γ-1)q²q^*_x + (γ-1)(γ-2)|q|⁴q = 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. 

Multi-Soliton Solutions for a Coupled System of the Nonlinear Schrödinger Equation and the Maxwell-Bloch Equations

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.