Solvability of a System of Nonlinear Wave Equations in Three-Dimensional Space

In this paper, we consider the Cauchy problem for a system of N∈ N coupled nonlinear wave equations in three-dimensional space. The system involves nonlinear terms that are monotone and satisfy concavity conditions. We first formulate the problem with prescribed initial data and then investigate the existence and uniqueness of positive, bounded, smooth solutions. To establish existence, we construct an iterative scheme that generates a sequence of approximate solutions. We prove that this sequence converges uniformly to a limit, which coincides with the solution of the original system. Furthermore, we show that under suitable structural assumptions on the nonlinearities, the solution obtained in this way is unique within the class under consideration. The results provide a constructive framework for analyzing coupled nonlinear wave equations and demonstrate applicability to a broad class of models arising in physics and engineering.