  The new perturbation theory for the problem of nonstationary anharmonic
oscillator with polynomial nonstationary perturbation is proposed. As a zero
order approximation the exact wave function of harmonic oscillator with
variable frequency in external field is used. Based on some intrinsic
properties of unperturbed wave function the variational-iterational method is
proposed, that make it possible to correct both the amplitude and the phase of
wave function. As an application the first order correction are proposed both
for wave function and S-matrix elements for asymmetric perturbation potential
of type $V(x,\tau)=\alpha (\tau)x^3+\beta (\tau)x^4.$ The transition amplitude
''ground state - ground state'' $W_{00}(\lambda ;\rho)$ is analyzed in detail
depending on perturbation parameter $\lambda $ (including strong coupling
region $% \lambda $ $\sim 1$) and one-dimensional refraction coefficient $\rho
$.
