On the Use of the Lie–Bäcklund Groups in the Context of Asymptotic Integrability

A new approach to the problem of asymptotic integrability of physical systems is developed and applied to the KdV equation with higher-order corrections. A central object of the approach is an integrable reference equation, which is constructed by defining a proper Lie– Bäcklund group of transformations and applying it to the leading order equation. It is shown that the solitary wave solutions of the asymptotically integrable equations, derived using asymptotic transformations, fail to approximate solutions of those equations in some (rather wide) range of the soliton and system parameters.