The new perturbation theory for the problem of the nonstationary anharmonic oscillator with polynomial nonstationary perturbation is proposed. As a zeroth-order approximation, the exact wavefunction of the harmonic oscillator with variable frequency in external field is used. Based on some intrinsic properties of unperturbed wavefunctions, the variational - iterational method is proposed, which makes it possible to correct both the amplitude and phase of the wavefunction. As an application, the first-order corrections are proposed for both the wavefunction and S-matrix elements for asymmetric perturbation potential of the type . The transition amplitude `ground state - ground state' is analysed in detail depending on the perturbation parameter (including the strong coupling region ) and the one-dimensional refraction coefficient .