I. Gapyak

This article discusses the structure of expansions that represent non-perturbative solutions of the Cauchy problem for the evolution equation hierarchies for the state and observables of many-particle systems with topological nearest-neighbor interaction. The generating operators for these expansions are derived using a proposed cluster expansion method applied to the groups of operators in the Liouville equations for both states and observables, respectively. The article also introduced the concept of a cumulant representation for distribution functions that describe the state of many particles with topological interactions and constructs a non-perturbative solution to the Cauchy problem for the hierarchy of nonlinear evolution equations for the cumulants of distribution functions. Furthermore, a relationship is established between the constructed solution and the series expansion structures for reduced distribution and correlation functions.