Systems of conservation laws with third-order Hamiltonian structures

Abstract

We investigate $n$-component systems of conservation laws that possess third-order Hamiltonian structures of differential-geometric type. The classification of such systems is reduced to the projective classification of linear congruences of lines in $\mathbb{P}^{n+2}$ satisfying additional geometric constraints. Algebraically, the problem can be reformulated as follows: for a vector space $W$ of dimension $n+2$, classify $n$-tuples of skew-symmetric 2-forms $A^α \in Λ^2(W)$ such that \[ ϕ_{βγ}A^β\wedge A^γ=0, \] for some non-degenerate symmetric $ϕ$.