INTEGRABILITY OF A CLASS OF SYSTEMS OF CONSERVATION-LAWS

One shows explicit integration formulae for the Cauchy problem partial derivative (t)u + partial derivative (x)f(u) = 0, u(x, 0) = u(o)(x). The allowed systems belong to the class formerly described by B. Temple, that is they have p families of invariant linear submanifolds. When p = n, the size of the system, the integration is complete, whereas if p < n, the integration reduces the system to a smaller one of n - p PDEs. For a scalar equation (n = 1), our formula reduces to well known calculus for Hamilton-Jacobi equations. The applications concern Chromatography, Electrophoresis and special linearly degenerate systems.