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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.

In this paper hyperbolic system of the first order gas dynamics PDE with initial and boundary conditions is studied. The aim of the paper is to construct the averaged system of differential equations in order to find the uniformly valid in a large domain asymptotical solution. The averaged system is a new object of asymptotical analysis.

Paper deals with the nonlinear coupled equations of the well known in the literature Hirota–Satsuma type system. The asymptotic analysis of this system, which is based on the principle of two scales and on averaging of weakly nonlinear hyperbolic systems along characteristics is presented in the paper. The asymptotic analysis shown that the system disintegrates on three independent Korteweg–de Vries equations in the non-resonance case, and the system describes an interaction of periodical nonlinear waves in the resonance case.

In this paper the model of the fluid flow in elastic pipe is studied. This model is described by the system of hyperbolic equations. The asymptotic analysis of these equations allows to investigate the conditions of the interaction of waves in the resonance case. The averaged system of integrodifferential equations is constructed to find the uniformly valid asymptotic solution in the long time interval and large interval of the dimensional variable.

The mathematical model of string nonlinear oscillations is presented. To found the asymptotic solution of the problem an averaging scheme was constructed in cited latest work of the second author. In this paper a methodology for construction special form soliutions of the system is proposed.

An algorithm for constructing approximations of averaging operators of quasi-periodical functions is presented in the paper. The algorithm can be used for constructinguniformly valid for a long time asymptotical solutions of differential equations with small parameter.