Simple waves and simple states in \(\mathbb{R}^2\) (Q1380373)

The paper deals with systems (1) \(\sum_{i,j} a^i_{jk}(x,u) \partial u^j/\partial x^i = b_k(x,u)\), where \(x = (x^1,\dots,x^n)\), \(u = (u^1,\dots,u^m)\). Sought are solutions \(u^j = u^j(R)\) that can be written in terms of a single function \(R(x)\), \(j = 1,\dots,m\). The starting point is an equivalent overdetermined system formed by equations (1) and \(\text{det}(\partial u^j/\partial x^i) = 0\). A practical solving procedure is suggested, based on the investigation of resolving distributions of Pfaff systems. Details are provided in the case \(m = n = 2\).