The author considers the Cauchy problem \[ u_{tt} - u^{2k} \sum^n_{i,j = 1} a_{ij} (t,x,u) u_{x_i x_j} = f(t,x,u,u_t), \quad u (0,x) = \Phi (x),\;u_t(0,x) = \Psi (x), \] where \(\Phi\), \(\Psi \in C_0^\infty (\mathbb{R}^n)\), \(k \in \mathbb{N}\), \(a_{ij} = a_{ji}\), \(f\) are \(C^\infty\)-functions, \(f(t,x,0,0) = 0\), and \[ \sum^n_{i,j = 1} a_{ij} \xi_i \xi_j \geq \lambda |\xi |^2 \] for some \(\lambda > 0\) and all \(\xi \in \mathbb{R}^n\). For this weakly hyperbolic system, a local existence and uniqueness result is proved for data satisfying \(\Phi \cdot \Psi \geq 0\). Then \(u\) is obtained as \(u(t, x) = \Phi (x) g(t,x) + \Psi (x) h(t,x)\) with \(C^\infty\)-functions \(g,h\). The proof is based on a combination of the Nash-Moser theorem with an extension of a result of \textit{O. A. Oleinik} [Commun. Pure Appl. Math. 23, 569-586 (1970; Zbl 0193.386)].