This work aims to obtain new transformations and auto-Bäcklund transformations for generalized Liouville equations with exponential nonlinearity having a factor depending on the first derivatives. This paper discusses the construction of Bäcklund transformations for nonlinear partial second-order derivatives of the soliton type with logarithmic nonlinearity and hyperbolic linear parts. The construction of transformations is based on the method proposed by Clairin for second-order equations of the Monge–Ampere type. For the equations studied in the article, using the Bäcklund transformations, new equations are found, which make it possible to find solutions to the original nonlinear equations and reveal the internal connections between various integrable equations.

The classical problem of stability of a pipeline section with fluid flow is considered in this paper. The equation of perturbed motion is solved by a method of expansion by forms of natural oscillations with further application of the Bubnov — Galerkin method. The boundary of the stability domain on the plane of fluid flow parameters is determined using the Raus — Hurwitz criterion for non-conservative stability problems. For fixed values of the relative mass, the trajectories of characteristic indicators are constructed as functions parametrically dependent on the velocity of the fluid flow. The frequency of pipeline oscillations in the event of loss of stability is determined by the flutter type. Flutter modes at various points of the boundary of the stability domain are examined. Flutter modes are represented by a beam of curved axes of the pipeline at discrete points of time throughout one period.

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