Systems of conservation laws in the context of the projective theory of congruences

S. I. Agafonov ; E. V. Ferapontov

Izvestiya. Mathematics , Tome 60 (1996) no. 6, pp. 1097-1122

Résumé

We associate to a system of $n$ conservation laws $$ u_t^i=f^i(u)_x, \qquad i=1,\dots,n, $$ an $n$-parameter family of lines in $(n+1)$-dimensional space $A^{n+1}$ given by the equations $$ y^i=u^iy^0-f^i(u), \qquad i=1,\dots,n. $$ Thereby we establish a correspondence between the reciprocal transformations of the system of conservation laws and the projective transformations of the space $A^{n+1}$, the rarefaction curves of the system of conservation laws and the developable surfaces of the associated family of lines, the Temple class of systems of conservation laws and the class of families of lines whose developable surfaces are either flat or conic. In the particular case $n=2$ the systems of the Temple class are explicitly described in terms of the theory of congruences.