Application of the averaging method to the investigation of nonlinear wave processes in elastic systems with circular symmetry (Q2784882)

The authors consider the nonlinear problem NEWLINE\[NEWLINE a_k''+ (\lambda_k^2-{\alpha'}_k^2)a_k = \varepsilon \Phi_k^{(1)} ({\bar a},{\bar a}', {\bar \alpha}, \sigma t), NEWLINE\]NEWLINE NEWLINE\[NEWLINE a_k\alpha_k''+ 2a_k'\alpha_k'=\varepsilon \Phi_k^{(2)}({\bar a}, {\bar a}', {\bar \alpha}, \sigma t),NEWLINE\]NEWLINE where \(a_k,\alpha_k\) are functions that characterize the amplitude and phase parameters of a wave process; \(\varepsilon \Phi_k^{(i)}, i=1,2\), are nonlinear analytic functions of vectors \( {\bar a}, {\bar a}', {\bar \alpha}.\) Moreover, the functions \( \varepsilon \Phi_k^{(i)}, i=1,2 \), are T-periodical, where \(T=\frac{2\pi}{\sigma};\) \( \varepsilon \) is a small parameter; \({\bar \lambda}=\{\lambda_1, \lambda_2, \ldots, \lambda_n\}\) is the frequency spectrum of the considered problem. The authors solve the problem of finding an approximate solution by means of an asymptotical averaging method.