The paper deals with the singularly perturbed Korteweg-de Vries equation with variable coefficients. The equation describes wave processes in various inhomogeneous media with variable characteristics and small dispersion. We consider the general algorithm of construction of asymptotic solutions of soliton type to the equation and present its approximate solutions of this type. We analyze properties of the constructed asymptotic solution depending on a small parameter. The results are demonstrated by the examples of the studied equation. We show that for an adequate description of qualitative properties of soliton type solutions to the singularly perturbed KdV equation with variable coefficients it is necessary to construct at least the first asymptotic approximation, that is, expansion containing both the main and the first term.
We study the influence of combined vibration perturbations on the vibration of a pipeline with flowing liquid. The study is carried out by using a finite-dimensional nonlinear model of a pipeline whose foundation suffers vibration perturbations in the longitudinal or transverse direction, or in both directions. We show how the combined perturbation of vibrations affects the dynamic stability of the system. The main attention is focused on the behavior of the system in a vicinity of the state where the rectilinear shape of the pipeline loses its stability.
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