Satoshi Tsujimoto

Casorati and Discrete Gram Type Determinant Representations of Solutions to the Discrete KP Hierarchy

Solutions to the discrete KP hierarchy are expressed by two types of determinants. The first expression is the Casorati determinant and the second one is the discrete Gram type determinant. In both cases, the difference formula for the determinants are given in a simple form and the bilinear equations of the hierarchy are reduced to the Laplace expansion for determinants.

Conserved Quantities of a Class of Nonlinear Difference-Difference Equations

Higher-order conserved quantities of a class of nonlinear difference-difference equations which are related to the discrete-time Toda equation are obtained using the Miura transformations.

Difference Scheme of Soliton Equations

The study of discrete-time integrable systems is currently the focus of an intense activity1, 2, 3, 4.

Pfaffian Representation of Solutions to the Discrete BKP Hierarchy in Bilinear Form

A discrete BKP hierarchy in bilinear form is given. It is shown that the solutions of the bilinear forms are given in terms of pfaffians and the bilinear equations of a discrete BKP hierarchy are reduced to an identity of pfaffians. The N -soliton solution is explicitly constructed in terms of the pfaffian. As an example of nonlinear difference equations, a difference analogue of the Sawada-Kotera equation is given from the discrete BKP hierarchy.

Ultradiscrete KdV Equation

We propose an ultradiscrete KdV equation in a coupled form and a Miura transformation relating it to the ultradiscrete Lotka-Volterra equation. Moreover we show that the ultradiscrete KdV equation under an appropriate boundary condition is equivalent to Takahashis soliton cellular automaton.

Discrete 2N-wave interaction

A difference scheme of a system of nonlinear partial differential equations describing 2N-wave interaction (9n = 1,2,....N,Uo=uN+1=0)Δ+xun(x,y)=un(x+δ,y)υn(x,y+e)−un(x,y)υn+1(x,y),Δ+yun(x,y)=υn(x,y+ϵ)un−1(x+δ,y)−υn(x,y)un(x,y), is obtained preserving an integrability of the system.