Valeriy Yumaguzhin

Invariants in relativity theory

In the relativistic theory the principle of naturality of physical laws leads to differential equations admitting diffeomorphisms of space-time as symmetries.In this paper we investigate differential invariants of actions of diffeomorphisms on (pseudo)- Riemann metrics, on solutions of the Einstein equation, and on solutions of the Einstein—Maxwell equation. We find the corresponding fields of rational differential invariants and the corresponding factor-equations.

On geometric structures of 2-dimensional gas dynamics equations

In this paper we describe geometric structures on solutions of the 2-dimensional gas dynamics equations. These structures are generated by characteristics of the system. We construct a 1-order tensor differential invariant of the structures. It is shown that there exists a unique linear connection on every solution of the gas dynamics equations such that its torsion coincides with the obtained invariant. We use this invariant to find classes of explicit torsion-free solutions of the gas dynamics equations for polytropic constant volume gas motion.

Minkowski metrics on solutions of the Khokhlov-Zabolotskaya equation

In this paper, we show that every solution of the Khokhlov-Zabolotskaya equation possesses a Minkowski metric. This metric is responsible for transport of singularities of KZ-solutions. We find explicit solutions for which the metrics are either locally-flat or Ricci-flat or conformally-flat.