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Abstract
The Hamiltonian formalism is used to derive the equations of long waves on the surface of an infinite layer of water over a horizontal, smooth bottom, taking into account second-order terms with respell to small parameters of non-linearity and dispersion; in other words, the Boussinesq equations of shallow-water theory are improved. A non-linear evolution equation is derived for the elevation of the free surface and a transformation is obtained to convert it into one of the higher-order Korteweg-de Vries (KdV) equations. Single- and double-soliton solutions are used to demonstrate the special features of the behaviour of the waves described by the equation, which are a more accurate version of KdV solitons.
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