Solution of nonlinear hyperbolic equations by an approximate analytical method

In this paper, we propose a method for solving the mixed problem for a hyperbolic equation with power nonlinearity. The first step of the method is reduction to the problem for the loaded equation containing the integral of a natural degree of the modulus of the unknown function. This integral expresses the norm of the unknown function in the corresponding Lebesgue space. Selection of constants of an a priori estimate allows us to linearize the loaded equation. A formula expressing the solution of the loaded equation by the solution of the ordinary differential equation associated with it is obtained. Approximation to the solution of the nonlinear equation is performed by means of an iterative process of solving a sequence of nonlinear problems.