On the connection of the Hamilton-Jacobi equation with some systems of quasilinear equations

We show that the Hamilton-Jacobi equation with some conditions on the Hamiltonian can be associated with a quasilinear system of equations of the first order, which can be reduced to the vector Hopf equation. We find relations between the system of Riemann invariants and a specially constructed Hamilton-Jacobi equation. The result is illustrated with examples of a system of isentropic gas dynamics equations and a system of equations of chromatography. It is shown that the method of stochastic perturbations along characteristics allows to associate with the Hamilton-Jacobi equation a system of conservation laws.