Tracy and Widom showed that the level spacing function of the Gaussian unitary ensemble is relate... more Tracy and Widom showed that the level spacing function of the Gaussian unitary ensemble is related to a particular solution of the fourth Painlevé equation. We reconsider this problem from the viewpoint of Hirota's bilinear method in soliton theory and present another proof. We also consider the asymptotic behavior of the level spacing function as s→∞, and its relation to the "Clarkson-McLeod solution" to the Painlevé IV equation.
We construct rational and piecewise-linear Yang-Baxter maps for a general N-reduction of the disc... more We construct rational and piecewise-linear Yang-Baxter maps for a general N-reduction of the discrete BKP equation.
Operators that intertwine representations of a degenerate version of the double affine Hecke alge... more Operators that intertwine representations of a degenerate version of the double affine Hecke algebra are introduced. Each of the representations is related to multi-variable orthogonal polynomials associated with Calogero-Sutherland type models. As applications, raising operators and shift operators for such polynomials are constructed.
The coupled KP hierarchy, introduced by Hirota and Ohta, are investigated by using the dressing m... more 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.
A generalized derivative nonlinear Schrödinger equation, q_t + q_xx + 2γ |q|^2 q_x + 2 (γ-1)q^2 q... more A generalized derivative nonlinear Schrödinger equation, q_t + q_xx + 2γ |q|^2 q_x + 2 (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.
We present a novel differential-difference system in (2+1)-dimensional space-time (one discrete, ... more 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 _2^tor.
A q-analogue of the gl_3 Drinfel'd-Sokolov hierarchy is proposed as a reduction of the q-KP h... more A q-analogue of the gl_3 Drinfel'd-Sokolov hierarchy is proposed as a reduction of the q-KP hierarchy. Applying a similarity reduction and a q-Laplace transformation to the hierarchy, one can obtain the q-Painleve VI equation proposed by Jimbo and Sakai.
The hierarchy structure associated with a (2+1)-dimensional Nonlinear Schroedinger equation is di... more The hierarchy structure associated with a (2+1)-dimensional Nonlinear Schroedinger 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.
We construct special solutions to the rational quantum Knizhnik-Zamolodchikov equation associated... more We construct special solutions to the rational quantum Knizhnik-Zamolodchikov equation associated with the Lie algebra gl_N. The main ingredient is a special class of the shifted non-symmetric Jack polynomials. It may be regarded as a shifted version of the singular polynomials studied by Dunkl. We prove that our solutions contain those obtained as a scaling limit of matrix elements of the vertex operators of level one.
We investigate common algebraic structure for the rational and trigonometric Calogero-Sutherland ... more 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.
We investigate algebraic structure for the BN-type Calogero model by using the exchange-operator ... more We investigate algebraic structure for the BN-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. 1
Scaling symmetry of ̂ gl n-type Drinfel’d-Sokolov hierarchy is investigated. Applying similarity ... more 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 Painlevé VI equation. We also discuss Weyl group symmetry of the hierarchy.
Tracy and Widom showed that the level spacing function of the Gaussian unitary ensemble is relate... more Tracy and Widom showed that the level spacing function of the Gaussian unitary ensemble is related to a particular solution of the fourth Painlevé equation. We reconsider this problem from the viewpoint of Hirota's bilinear method in soliton theory and present another proof. We also consider the asymptotic behavior of the level spacing function as s→∞, and its relation to the "Clarkson-McLeod solution" to the Painlevé IV equation.
We construct rational and piecewise-linear Yang-Baxter maps for a general N-reduction of the disc... more We construct rational and piecewise-linear Yang-Baxter maps for a general N-reduction of the discrete BKP equation.
Operators that intertwine representations of a degenerate version of the double affine Hecke alge... more Operators that intertwine representations of a degenerate version of the double affine Hecke algebra are introduced. Each of the representations is related to multi-variable orthogonal polynomials associated with Calogero-Sutherland type models. As applications, raising operators and shift operators for such polynomials are constructed.
The coupled KP hierarchy, introduced by Hirota and Ohta, are investigated by using the dressing m... more 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.
A generalized derivative nonlinear Schrödinger equation, q_t + q_xx + 2γ |q|^2 q_x + 2 (γ-1)q^2 q... more A generalized derivative nonlinear Schrödinger equation, q_t + q_xx + 2γ |q|^2 q_x + 2 (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.
We present a novel differential-difference system in (2+1)-dimensional space-time (one discrete, ... more 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 _2^tor.
A q-analogue of the gl_3 Drinfel'd-Sokolov hierarchy is proposed as a reduction of the q-KP h... more A q-analogue of the gl_3 Drinfel'd-Sokolov hierarchy is proposed as a reduction of the q-KP hierarchy. Applying a similarity reduction and a q-Laplace transformation to the hierarchy, one can obtain the q-Painleve VI equation proposed by Jimbo and Sakai.
The hierarchy structure associated with a (2+1)-dimensional Nonlinear Schroedinger equation is di... more The hierarchy structure associated with a (2+1)-dimensional Nonlinear Schroedinger 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.
We construct special solutions to the rational quantum Knizhnik-Zamolodchikov equation associated... more We construct special solutions to the rational quantum Knizhnik-Zamolodchikov equation associated with the Lie algebra gl_N. The main ingredient is a special class of the shifted non-symmetric Jack polynomials. It may be regarded as a shifted version of the singular polynomials studied by Dunkl. We prove that our solutions contain those obtained as a scaling limit of matrix elements of the vertex operators of level one.
We investigate common algebraic structure for the rational and trigonometric Calogero-Sutherland ... more 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.
We investigate algebraic structure for the BN-type Calogero model by using the exchange-operator ... more We investigate algebraic structure for the BN-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. 1
Scaling symmetry of ̂ gl n-type Drinfel’d-Sokolov hierarchy is investigated. Applying similarity ... more 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 Painlevé VI equation. We also discuss Weyl group symmetry of the hierarchy.
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Papers by Saburo Kakei