Group analysis of hydrodynamic-type systems

TL;DR

This paper investigates symmetries and linearizations of hydrodynamic-type systems, deriving conditions for infinite-dimensional symmetry groups, constructing recursion operators, and finding exact solutions, with a focus on two-component Hamiltonian and semi-Hamiltonian systems.

Contribution

It establishes existence conditions for infinite-dimensional symmetry groups and derives linearizing transformations for hydrodynamic-type systems, advancing understanding of their symmetry structures.

Findings

Derived linearizing transformations for hydrodynamic systems

Constructed recursion operators for symmetry generation

Analyzed symmetry conditions for Hamiltonian and semi-Hamiltonian systems

Abstract

We study point and higher symmetries for the hydrodynamic-type systems with two independent variables $t$ and $x$ with and without explicit dependence of the equations on $t,x$. We consider those systems which possess an infinite-dimensional group of the hydrodynamic symmetries, establish existence conditions for this property and, using it, derive linearizing transformations for these systems. The recursion operators for symmetries are obtained and used for constructing infinite series of exact solutions of the studied equations. Higher symmetries, i.e. the Lie-Backlund transformation groups, are also studied and the interrelation between the existence conditions for higher symmetries and recursion operators is established. More complete results are obtained for two-component systems, though $n$-component systems are also studied. In particular, we consider Hamiltonian and semi-Hamiltonian systems.