The evolution of a solitary wave with very weak nonlinearity which was
originally investigated by Miles [4] is revisited. The solution for a
one-dimensional gravity wave in a water of uniform depth is considered. This
leads to finding the solution to a Korteweg-de Vries (KdV) equation in which
the nonlinear term is small. Also considered is the asymptotic solution of the
linearized KdV equation both analytically and numerically. As in Miles [4], the
asymptotic solution of the KdV equation for both linear and weakly nonlinear
case is found using the method of inversescattering theory. Additionally
investigated is the analytical solution of viscous-KdV equation which reveals
the formation of the Peregrine soliton that decays to the initial sech^2(\xi)
soliton and eventually growing back to a narrower and higher amplitude
bifurcated Peregrine-type soliton.Comment: 15 page
