On the asymptotic solutions of the KdV equation with higher-order corrections

A method for construction of new integrable PDEs, whose properties are related to an asymptotic perturbation expansion with the leading-order term given by an integrable equation, is developed. A new integrable equation is constructed by applying the properly defined Lie-Bäcklund group of transformations to the leading-order equation. The integrable equations related to the Korteweg-de Vries (KdV) equation with higher-order corrections are used to investigate the limits of applicability of the so-called asymptotic integrability concept. It is found that the solutions of the higher-order KdV equations obtained by a near identity transform from the normal form solitary waves cannot, in principle, describe some intrinsic features of the high-order KdV solitons.