Cremona transformation and general solution of one dynamical system of the static model

Abstract

The general solution to equations for S-matrix elements of the three-channel static model with the crossing-symmetry matrix A(1, 1) in their famous formulation as a system of nonlinear difference equations is obtained for the first time. This three-dimensional dynamical system is defined essentially by the two-dimensional subsystem given by the mapping that is the quadratic Cremona transformation. For the latter all invariant algebraic curves are found. On their basis, the Cremona transformation is constructed, which transforms the subsystem to a very simple form of the mapping with a zero fixed hyperbolic point and a Jacobian not equal to one, which enables one to find an invariant nonalgebraic curve and to investigate its properties. We have found the invariant measure and Cremona transformation reducing the subsystem to the form of area-preserving mapping. Using the results of Birkhoff and Moser for the mapping we transform the initial difference equations into an easily integrable form and obtain two types of the general solution. The phase curves of the subsystem for different values of the invariant of the Cremona transformation are computed.