Abstract
A weakly non-linear oscillatory system with distributed parameters is investigated An asymptotic method of constructing a solution, which describes the oscillatory motions of the single-mode (single-frequency) approximation, which is usually implemented in practical problems, is described and justified. Constructive sufficient conditions are formulated and the closeness of the approximate single-frequency solution to the exact solution in an asymptotically long time interval is proved. Possible extensions of the structure of the perturbing functions are considered and the case of the finite-mode approximation is investigated. Solutions of specific problems, which are of practical interest, are constructed to illustrate the effectiveness of the single-frequency approximation method.
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