On the evolutions of the discontinuities of orders \(\geq m\) in solutions for quasi-linear systems of PDEs of order m, or for the linarizations of these systems (Q1820923)

One considers a system \(L[u]=0\) of PDEs, quasi-linear and of order m, which possesses a bicharacteristic line \(\ell\), as it happens in the hyperbolic case. For \(\nu =0,...,\mu -m(>0)\) let \(u^{(\nu)}\) be a discontinuity wave of order \(m+\nu\) that solves the system above and whose discontinuity hypersurface includes \(\ell\). The corresponding transport equations along \(\ell\) are considered. Furthermore some interesting cases are pointed out, in which these equations turn out to be mutually equivalent in a suitable sense. Some theorems are stated to compare the transport equations for the discontinuities of the above kinds, that are connected with the systems \(d^ hL[u]/dt^ h=0\) \((h=0,...,\mu -m)\) and/or the linearization of the system \(L[u]=0\) around any regular solution of it.