Substantiation of asymptotical solution of weakly nonlinear hyperbolic system

A hyperbolic system of first order partial differential equations with small parameter and periodical initial conditions is obtained. The uniformly valid in a long time interval asymptotical solution can be constructed using the averaging along characteristics of the system. The asymptotical estimation of the approximation is made in the paper. Some aspects of practical realization of the method are discussed in the work too.