Comment on "Shallow-water soliton dynamics beyond the Korteweg-de Vries equation"

The authors of the paper "Shallow-water soliton dynamics beyond the Korteweg - de Vries equation" [1] write that they derived a new nonlinear equation describing shallow water gravity waves for an uneven bottom in the form of the higher (fifth)-order Korteweg - de Vries equation for surface elevation. The equation has been obtained by applying a perturbation method [2] for specific relations between the orders of the three small parameters of the problem α=O(β) and δ=O(β) up to the second order in β. In this comment, it is shown that the derivation presented in [1] is inconsistent because of an oversight concerning the orders of terms in equations of the Boussinesq system. Therefore the results, in particular, the new evolution equation and the dynamics that it describes, bear no relation to the problem under consideration. A consistent derivation is presented, and also results of applying the perturbation procedure with some other orderings between the small parameters are given to provide a broader view of the problem. Several new nonlinear evolution equations governing small amplitude shallow water waves for an uneven bottom have been derived.