General solutions of the nonlinear wave equation and of the eikonal equation (Q1185172)

The overdetermined system of wave and eikonal equations \(\partial^ 2u/\partial t^ 2-\Delta u=F_ 1(u)\), \((\partial u/\partial t)^ 2- \text{grad} u\cdot\text{grad} u=F_ 2(u)\) is considered for \(x\in\mathbb{R}^ 3\) in the cases \(F_ 1=F_ 2=0\) and \(F_ 2=1\). Necessary and sufficient conditions for compatibility are proved and general analytic description of smooth solutions is given by means of certain constructive method of integration.