Classification of integrable Vlasov-type equations

TL;DR

This paper classifies integrable Vlasov-type equations by solving a key functional equation, providing a general solution expressed through hypergeometric functions, advancing understanding of their integrability conditions.

Contribution

It introduces a novel classification approach by reducing the problem to a functional equation and solving it explicitly with hypergeometric functions.

Findings

Derived a functional equation for the generating function.

Obtained a general solution in terms of hypergeometric functions.

Provided a classification framework for integrable Vlasov-type equations.

Abstract

Classification of integrable Vlasov-type equations is reduced to a functional equation for a generating function. A general solution of this functional equation is found in terms of hypergeometric functions.