Classification of conservative hydrodynamic chains. Vlasov type kinetic equation, Riemann mapping and the method of symmetric hydrodynamic reductions

TL;DR

This paper provides a complete classification of integrable conservative hydrodynamic chains using special coordinates, expressing coefficients via modular forms and hypergeometric functions, and introduces related Lax representation, Vlasov equation, and Riemann mapping.

Contribution

It introduces a comprehensive classification of integrable hydrodynamic chains with explicit coefficient expressions and associated Lax, Vlasov, and Riemann mapping structures.

Findings

Explicit classification of integrable hydrodynamic chains.

Derivation of a dispersionless Lax representation.

Construction of a Riemann mapping and conservation laws.

Abstract

A complete classification of integrable conservative hydrodynamic chains is presented. These hydrodynamic chains are written via special coordinates -- moments, such that right hand sides of these infinite component systems depend linearly on a discrete independent variable $k$. All variable coefficients of these hydrodynamic chains can be expressed via modular forms with respect to moment $A^{0}$, via hypergeometric functions with respect to moment $A^{1}$; they depend polynomially on moment $A^{2}$ and linearly on all other higher moments $A^{k}$. A dispersionless Lax representation is found. Corresponding collisionless Boltzmann (Vlasov like kinetic) equation is derived. A Riemann mapping is constructed. A generating function of conservation laws and commuting flows is presented.