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An amplitude equation for long, nonlinear internal waves in a weak shear flow over topography

A forced, variable coefficients Kor\-te\-weg-de Vries equation for amplitudes of long, nonlinear internal waves in a stratified shear flow over topography is derived when the magnitude of the basic flow is small. The derivation is done by incorporating the basic flow in the perturbation series of the Taniuti and Wei's reductive perturbation method. Istead of the long-wave Tailor-Goldstein spectral problem, only the usual long-wave spectral problem for rested stratified media and a boundary value problem have to be solved for calculating the coefficients of obtained equation. Explicit expressions for these coefficients are presented.