Jun Mada

Irreducibility and co-primeness as an integrability criterion for discrete equations

We study the Laurent property, the irreducibility and co-primeness of discrete integrable and non-integrable equations. First we study a discrete integrable equation related to the Somos-4 sequence, and also a non-integrable equation as a comparison. We prove that the conditions of irreducibility and co-primeness hold only in the integrable case. Next, we generalize our previous results on the singularities of the discrete Korteweg–de Vries (dKdV) equation. In our previous paper (Kanki et al 2014 J. Phys. A: Math. Theor. 47 065201) we described the singularity confinement test (one of the integrability criteria) using the Laurent property, and the irreducibility, and co-primeness of the terms in the bilinear dKdV equation, in which we only considered simplified boundary conditions. This restriction was needed to obtain simple (monomial) relations between the bilinear form and the nonlinear form of the dKdV equation. In this paper, we prove the co-primeness of the terms in the nonlinear dKdV equation for general initial conditions and boundary conditions, by using the localization of Laurent rings and the interchange of the axes. We assert that co-primeness of the terms can be used as a new integrability criterion, which is a mathematical re-interpretation of the confinement of singularities in the case of discrete equations.

On the initial value problem of a periodic box-ball system

We show that the initial value problem of a periodic box-ball system can be solved in an elementary way using simple combinatorial methods.

Discrete Integrable Equations over Finite Fields

Discrete integrable equations over finite fields are investigated. The indeter- minacy of the equation is resolved by treating it over a field of rational functions instead of the finite field itself. The main discussion concerns a generalized discrete KdV equation related to a Yang{Baxter map. Explicit forms of soliton solutions and their periods over finite fields are obtained. Relation to the singularity confinement method is also discussed.

Discrete Painlevé II equation over finite fields

We investigate the discrete Painleve II equation over finite fields. We treat it over local fields and observe that it has a property that is similar to the good reduction over finite fields. We can use this property, which seems to be an arithmetic analogue of singularity confinement, to avoid the indeterminacy of the equations over finite fields and to obtain special solutions from those defined originally over fields of characteristic zero.

Path description of conserved quantities of generalized periodic box-ball systems

We investigate conserved quantities of periodic box-ball systems (PBBS) with arbitrary kinds of balls and box capacity greater than or equal to 1. We introduce the notion of nonintersecting paths on the two dimensional array of boxes, and give a combinatorial formula for the conserved quantities of the generalized PBBS using these paths.

The exact correspondence between conserved quantities of a periodic box-ball system and string solutions of the Bethe ansatz equations

We investigate the link between a periodic box-ball system (PBBS) and a solvable lattice model. Introducing a PBBS with an integer parameter corresponding to the dimensionality of the auxiliary space for the lattice model, we prove an important relationship between the conserved quantities of states of the PBBS and eigenvectors constructed through the string hypothesis.

Conserved quantities and generalized solutions of the ultradiscrete KdV equation

We construct generalized solutions to the ultradiscrete KdV equation, including the so-called negative solition solutions. The method is based on the ultradiscretization of soliton solutions to the discrete KdV equation with gauge transformation. The conserved quantities of the ultradiscrete KdV equation are shown to be constructed in a similar way to those for the box–ball system.

Asymptotic behaviour of fundamental cycle of periodic box-ball systems

We investigate asymptotic behaviour of fundamental cycle of periodic box–ball systems (PBBSs) when the system size N goes to infinity. According to integrable nature of the PBBS, the trajectory is confined to qualitatively smaller number of states than that of the total states. We prove that, although the maximum fundamental cycle is of order of exp[√N], almost all fundamental cycle is less than exp[(logN)2].

Integrability criterion in terms of coprime property for the discrete Toda equation

We reformulate the singularity confinement of the discrete Toda equation. We prove the co-primeness property, which has been introduced in our previous paper (arXiv:1311.0060) as one of the integrability criteria, for the discrete Toda equation. We study three types of boundary conditions (semi-infinite, molecule, periodic) for the discrete Toda equation, and prove that the same co-primeness property holds for all the types of boundaries. (v2: typos corrected, final version to appear in J. Math. Phys.)

Fundamental cycle of a periodic box–ball system and solvable lattice models

We investigate the fundamental cycle of a periodic box–ball system (PBBS) from a relation between the PBBS and a solvable lattice model. We show that the fundamental cycle of the PBBS is obtained from eigenvalues of the transfer matrix of the solvable lattice model.