Multiple scales analysis of the steady-state Korteweg-de Vries equation perturbed by a damping term

(i) For laminar flow (1) has been derived in [7] for f = −H ′′. A first analysis of that case is in [6]. Moreover this equation is also considered in the context of shock profiles in plasmas [2]. (ii) For turbulent flow in a certain distinguished limit concerning the Reynolds number and the Froude number eq. (1) has been derived in [3,4] with f = H . A multiple scales analysis of the full flow problem can be found in [11]. That equation has been previously mentioned by [10] in context of a resonant flow of a stratified fluid over a topography.