Evolution of perturbations on a weakly inhomogeneous background

The evolution of growing and decaying one-dimensional linear perturbations on a stationary, weakly inhomogeneous background is investigated studied. Attention is focused on the amplification of waves that arise from initial perturbations, localized in regions whose width is small compared with the inhomogeneity scale. A relation between the Hamiltonian formalism (with a complex dispersion equation) and the saddle-point method is established for an asymptotic representation of the integral that expresses perturbations in terms of the initial data. Model examples of the evolution of perturbations are examined.