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We consider a generalization of the projecting operators method for the caseof Cauchy problem for systems of 1D evolution differential equations of firstorder with variable coefficients. It is supposed that the coefficientsdependence on the only variable x is weak, that is described by a smallparameter introduction. Such problem corresponds, for example, to the case ofwave propagation in a weakly inhomogeneous medium. As an example, we specifythe problem to adiabatic acoustics. For the Cauchy problem, to fixunidirectional modes, the projection operators are constructed. The method ofsuccessive approximations (perturbation theory) is developed and based onpseudodifferential operators theory. The application of these projectionoperators allows to obtain approximate evolution equations corresponding to theseparated directed waves.

The method of dynamic projection operators in the theory of hyperbolic systems of partial differential equations with variable coefficients