A Characterization of Discrete Time Soliton Equations

A discretization of independent variables of integrable nonlinear differential equations breakes their integrability in general. Therefore it is remarkable that there exist certain discrete analogue of integrable differential equations which preserve integrability[2][16]. Integrability of ODEs can be tested by studying whether exist singularities which depend on initial values, a method called Painlevé test. Therefore it is natural to look for a discrete analogue of the Painlevé property which enables one to predict behaviour, deterministic or chaotic, of a sequence of map starting from some initial value. There has been, however, not known a discrete analogue of Painlevé test despite of some useful proposals[17]. In our previous paper we have studied a discrete analogue of the Lotka-Volterra equation which is known completely integrable[18]. Considering it as a sequence of a map it was observed that at every step of the map there are two possibilities to be chosen. Hence the orbit is not deterministic. Nevertheless the map is compatible with integrability of the equation. This happens due to the fact that one of two types of the map, which we call B-type, does not generate a new orbit but simply exchanges two different orbits created by the other type (A-type) of the map. Starting from a symmetric discrete Lotka-Volterra equation (dLV) under the periodic boundary conditions the map is determined solving a quadratic equation. The expression under the square root which appears in the solutions turns out to be a perfect square in this problem, hence no singularity arises in the map. Out of the two maps the A-type