Saburo Kakei

Bilinearization of a generalized derivative nonlinear Schrödinger equation

A generalized derivative nonlinear Schrodinger equation, i q t + q x x +2 iγ∣ q ∣ 2 q x +2 i(γ- 1) q 2 q x * +(γ-1)(γ-2)∣ q ∣ 4 q =0, is studied by means of Hirotas bilinear formalism. Soliton solutions are constructed as quotients of Wronski-type determinants. A relationship between the bilinear structure and gauge transformation is also discussed.

Orthogonal and symplectic matrix integrals and coupled KP hierarchy

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.

Yang-Baxter maps and the discrete KP hierarchy

We present a systematic construction of the discrete KP hierarchy in terms of Sato–Wilson-type shift operators. Reductions of the equations in this hierarchy to 1+1-dimensional integrable lattice systems are considered, and the problems that arise with regard to the symmetry algebra underlying the reduced systems as well as the ultradiscretizability of these systems are discussed. A scheme for constructing ultradiscretizable reductions that give rise to Yang–Baxter maps is explained in two explicit examples.

Similarity reduction of the modified Yajima–Oikawa equation

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 A(1)2. The system of equations for self-similar solutions is presented as a Hamiltonian system of degree of freedom 2, and admits a group of Backlund transformations isomorphic to the affine Weyl group of type A(1)2. We show that the system is equivalent to a two-parameter family of the fifth Painleve equation.

The Rational qKZ Equation and Shifted Non-Symmetric Jack Polynomials ⋆

We construct special solutions to the rational quantum Knizhnik-Zamolodchikov equation associated with the Lie algebra glN. 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.

Yang-Baxter Maps from the Discrete BKP Equation ?

We construct rational and piecewise-linear Yang-Baxter maps for a general N-reduction of the discrete BKP equation.

TOROIDAL LIE ALGEBRA AND BILINEAR IDENTITY OF THE SELF-DUAL YANG-MILLS HIERARCHY

Bilinear identity associated with the self-dual Yang-Mills hierarchy is discussed by using a fermionic representation of the toroidal Lie algebra sltor 2 .