On application of Fourier method for one class of equations describing nonlinear oscillations

Solutions of nonlinear partial differential equations with a small parameter are constructed as a sum of truncated Fourier series and some additional function. The coefficients of the truncated Fourier series depend on the small parameter and satisfy a nonlinear system of ordinary differential equations, which in turn is determined by nonlinear partial differential equation. A certain class of equations describing nonlinear oscillations was determined, for which the coefficients of the truncated Fourier series are bounded functions of time. This fact makes it possible to estimate the additional function and to justify the applicability of the Fourier method for the constructed class of nonlinear partial differential equations.