Linear Congruences and hyperbolic Systems of conservation Laws

TL;DR

This paper explores the connection between hyperbolic systems of conservation laws and congruences of lines in projective space, using algebraic geometry to classify certain systems and their geometric representations.

Contribution

It introduces a geometric framework linking conservation laws to line congruences and classifies linear congruences in projective 5-space related to Temple systems.

Findings

Classification of linear congruences in P^5.

Correspondence between hyperbolic systems and line congruences.

Application of algebraic geometry to conservation laws.

Abstract

S. I. Agafonov and E. V. Ferapontov have introduced a construction that allows naturally associating to a system of partial differential equations of conservation laws a congruence of lines in an appropriate projective space. In particular hyperbolic systems of Temple class correspond to congruences of lines that place in planar pencils of lines. The language of Algebraic Geometry turns out to be very natural in the study of these systems. In this article, after recalling the definition and the basic facts on congruences of lines, Agafonov-Ferapontov's construction is illustrated and some results of classification for Temple systems are presented. In particular, we obtain the classification of linear congruences in $\mathbb{P}^5$, which correspond to some classes of $T$-systems in 4 variables.