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 [1] 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 reinterpretation of the confinement of singularities in the case of discrete equations.
We investigate some of the discrete Painlevé equations (dP II , qP I and qP II ) and the discrete KdV equation over finite fields 1 . The first part concerns the discrete Painlevé equations. We review some of the ideas introduced in our previous papers [1,2] and give some detailed discussions. We first show that they are well defined by extending the domain according to the theory of the space of initial conditions. We then extend them to the field of p-adic numbers and observe that they have a property that is called an 'almost good reduction' of dynamical systems over finite fields. We can use this property, which can be interpreted as 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. In the second part we study the discrete KdV equation. We review the discussions in [3] and present a way to resolve the indeterminacy of the equation by treating it over a field of rational functions instead of the finite field itself. Explicit forms of soliton solutions and their periods over finite fields are obtained.
We study the distribution of singularities for partial difference equations, in particular, the bilinear and nonlinear form of the discrete version of the Korteweg–de Vries (dKdV) equation. Using the Laurent property, and the irreducibility, and co-primeness of the terms of the bilinear dKdV equation, we clarify the relationship of these properties with the appearance of zeros in the time evolution. The results are applied to the nonlinear dKdV equation and we formulate the famous integrability criterion (singularity confinement test) for nonlinear partial difference equations with respect to the co-primeness of the terms.
We show that the initial value problem of a periodic box-ball system can be solved in an elementary way using simple combinatorial methods.A periodic box-ball system (PBBS) is a dynamical system of balls in an array of boxes with a periodic boundary condition [1,2]. The PBBS is obtained from the discrete KdV equation and the discrete Toda equation, both of which are known as typical integrable nonlinear discrete equations, through a limiting procedure called ultradiscretization [3,4]. Since the ultradiscretization preserves the main properties of the original discrete equations, and the solvability of the initial value problem being an important property of integrable equations, we expect that the initial value problem of the PBBS can also be solved. In fact, the initial value problem for the PBBS was first solved by inverse ultradiscretization combined with the method of inverse scattering transform of the discrete Toda equation [5] and recently by the Bethe ansatz for an integrable lattice model with quantum group symmetry at the deformation parameter q = 0 and q = 1 [6]. These two methods, however, require fairly specialized mathematical knowledge on algebraic curves or representation theory of quantum algebras.An important property which characterizes a state of the PBBS is the fundamental cycle of the state, i.e., the length of the trajectory to which it belongs. Its explicit formula as well as statistical distribution was obtained and its relation to the celebrated Riemann hypothesis was clarified [7,8,9]. To prove the formula for fundamental cycle, one of the key steps is to compare a state with its 'reduced states' constructed by the '10-elimination'. In this article, we show that the initial value problem of the PBBS is solved by simple combinatorial arguments -essentially given in Ref.[7] -with some remarkable features of the reduced states.First we quickly review the definition of the PBBS and its conserved quantities. Consider a one-dimensional array of boxes each with a capacity of one ball. A periodic boundary condition is imposed by assuming that the last box is adjacent to the first one. Let the number of boxes be N and that of balls be M . We assume M < N/2. An arrangement of M balls in N boxes is called a state of the PBBS. Denoting a vacant box by 0 and a filled box by 1, a state 1
Any state of the box-ball system (BBS) together with its time evolution is described by the N-soliton solution (with appropriate choice of N) of the ultradiscrete KdV equation. It is shown that simultaneous elimination of all '10'-walls in a state of the BBS corresponds exactly to reducing the parameters that determine 'the size of a soliton' by one. This observation leads to an expression for the solution to the initial-value problem (IVP) for the BBS. Expressions for the solution to the IVP for the ultradiscrete Toda molecule equation and the periodic BBS are also presented.
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