Theory of congruences and systems of conservation laws
This paper focusses on new formulations of systems of conservation laws by using the geometrical language of the theory of congruence. The central part of the paper provides a correspondence between a system of conservation laws \[ u^i_t= (f^i({\mathbf u}))_x,\quad {\mathbf u}=(u^1,u^2,\dots ,u^n),\qquad i=1,\dots,n \] and an \(n\)-parameter family of straight lines in the \((n+1)\)-dimensional space with coordinates \(y^0,y^1,\dots,y^n\) defined by the equations \[ y^i=u^iy^0-f^i({\mathbf u}),\quad i=1,\cdots,n. \] The reciprocal transformations, the rarefaction curves, the shock adiabatic curves and the commuting property of systems of conservation laws are interpreted in the language of the theory of congruence. The Temple class of systems of conservation laws, especially with two components [\textit{B. Temple}, Trans. Am. Math. Soc. 280, 781-795 (1983; Zbl 0559.35046)], is analyzed in detail.
scientific article; zbMATH DE number 1348489
scientific article