An averaging method for a quasilinear hyperbolic system. Asymptotics of solutions

In a multidimensional space-time strip, the Cauchy problem is considered for a quasilinear hyperbolic system of first-order differential equations with rapidly oscillating time data that do not explicitly depend on spatial variables. The Krylov–Bogolyubov averaging method is substantiated for it, and an algorithm for constructing complete asymptotics of solutions based on this method and the method of two-scale expansions is substantiated. The results obtained are directly applicable to the generalized Hopf equation with rapidly oscillating data.