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

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.

References in corpus (3)

• Classification of integrable hydrodynamic chains and generating functions of conservation laws
• The Hamiltonian approach in classification and integrability of hydrodynamic chains
• Differential Geometry of Hydrodynamic Vlasov Equations