On Integration of the Nonlinear d’Alembert- Eikonal System and Conditional Symmetry of Nonlinear Wave Equations

We study integrability of a system of nonlinear partial differential equations consisting of the nonlinear d’Alembert equation 2u = F (u) and nonlinear eikonal equation uxμuxμ = G(u) in the complex Minkowski space R(1, 3). A method suggested makes it possible to establish necessary and sufficient compatibility conditions and construct a general solution of the d’Alembert-eikonal system for all cases when it is compatible. The results obtained can be applied, in particular, to construct principally new (non-Lie, non-similarity) solutions of the non-linear d’Alembert, Dirac, and YangMills equations. Solutions found in this way are shown to correspond to conditional symmetry of the equations enumerated above. Using the said approach, we study in detail conditional symmetry of the nonlinear wave equation 2w = F0(w) in the fourdimensional Minkowski space. A number of new (non-Lie) reductions of the above equation are obtained giving rise to its new exact solutions which contain arbitrary functions.