Shin Isojima

Singularity confinement test for ultradiscrete equations with parity variables

We present a new ultradiscretization method which does not require that the solutions of the discrete equation have a fixed sign. We construct an ultradiscrete analogue of the singularity confinement test using this method and thereby propose an integrability test for ultradiscrete equations.

A Class of Special Solutions for the Ultradiscrete Painlevé II Equation

A class of special solutions are constructed in an intuitive way for the ultradis- crete analog of q-Painleve II (q-PII) equation. The solutions are classified into four groups depending on the function-type and the system parameter.

Direct ultradiscretization of Ai and Bi functions and special solutions for the Painlevé II equation

Ultradiscrete Ai and Bi functions are directly derived through the ultradiscrete limit from q-difference analogues of the Ai and Bi functions, respectively. An infinite number of identities among the number of restricted partitions are obtained as by-products. A direct relationship between a class of special solutions for the ultradiscrete Painlev? II equation and those of the q-Painlev? II equation which have a determinantal structure is also established.

New Airy-type solutions of the ultradiscrete Painlevé II equation with parity variables

The q-difference Painleve II equation admits special solutions written in terms of determinant whose entries are the general solution of the q-Airy equation. An ultradiscrete limit of the special solutions is studied by the procedure of ultradiscretization with parity varialbes. Then we obtain new Airy-type solutions of the ultradiscrete Painleve II equation with parity variables, and the solutions have richer structure than the known solutions.

Do ultradiscrete systems with parity variables satisfy the singularity confinement criterion

Ultradiscrete singularity confinement test, which is an integrability detector for ultradiscrete equations with parity variables, is applied to various ultradiscrete equations. The ultradiscrete equations exhibit singularity structures analogous to those of the discrete counterparts. Exact solutions to linearisable ultradiscrete equations are also constructed to explain the singularity structures.

Discrete and ultradiscrete Bäcklund transformation for KdV equation

The Backlund transformation for the discrete Korteweg–de Vries equation is introduced in the bilinear form. The superposition formula is also derived from the transformation. An ultradiscrete analogue of the transformation is presented by means of the ultradiscretization technique. This analogue gives the Backlund transformation for the box and ball system. The ultradiscrete soliton solutions for the system are also discussed with explicit examples.

On exact solutions with periodic structure of the ultradiscrete Toda equation with parity variables

Exact solutions of the ultradiscrete Toda equation with parity variables are constructed from the soliton solutions of the discrete Toda lattice equation. The solution has a periodic phase constant and describes a traveling pulse with a periodic variation. Its behavior is different from that of the usual soliton solution.

Tropical Krichever construction for the non-periodic box and ball system

A solution for an initial value problem of the box and ball system is constructed from a solution of the periodic box and ball system. The construction is done through a specific limiting process based on the theory of tropical geometry. This method gives a tropical analogue of the Krichever construction, which is an algebro-geometric method to construct exact solutions to integrable systems, for the non-periodic system.