Abstract
The case of minimal energy damping of small-amplitude dispersive shallow-water waves is considered for a viscous incompressible fluid described by the Navier-Stokes equation. Using variational methods, a fourth-order linear differential equation for the stream function is derived. From this equation and the appropriate boundary conditions an approximate damping rate for a soliton propagating on the surface of the fluid is calculated by matching the velocity on the upper surface to the velocity on the upper surface obtained from the Korteweg-de Vries equation and by calculating the time rate of change of the horizontal momentum of the fluid.
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