Lotka–Volterra systems satisfying a strong Painlevé property

Abstract

Abstract We use a strong version of the Painlevé property to discover and characterize a new class of n-dimensional Hamiltonian Lotka–Volterra systems, which turn out to be Liouville integrable as well as superintegrable. These systems are in fact Nambu systems, they posses Lax equations and they can be explicitly integrated in terms of elementary functions. We apply our analysis to systems containing only quadratic nonlinearities of the form aijxixj,i≠j, and require that all variables diverge as t−1. We also require that the leading terms depend on n−2 free parameters. We thus discover a cocycle relation among the coefficients aij of the equations of motion and by integrating the cocycle equations we show that they are equivalent to the above strong version of the Painlevé property. We also show that these systems remain explicitly solvable even if a linear term bixi is added to the i-th equation, even though this violates the Painlevé property, as logarithmic singularities are introduced in the Laurent solutions, at the first terms following the leading order pole.