On the use of the Lie-B\"acklund groups in the context of asymptotic integrability

Abstract:
For many important physical systems, the leading order term in an asymptotic perturbation expansion is given by an integrable nonlinear equation. The meaning of the term "asymptotic integrability" is that the equation with higher order corrections is integrable up to a certain order in the asymptotic sense. In the present paper, we develop the approach to defining conditions for asymptotic integrability of physical systems which differs conceptually from that of the normal form theory. The central point of the approach is some reference integrable equation which is constructed by applying the Lie-B\"acklund group of transformations to the leading order equation. In general, the transformations and the reference equation are  represented by the Lie series, and conditions for asymptotic integrability are defined by relating the higher order terms in an asymptotic perturbation expansion for the physical system with the corresponding terms of an expansion of the reference equation. In particular cases, when the Lie-B\"acklund equations can be solved in a closed form, the reference equation is explicitly defined. This new integrable equation, which is of the same order as the leading order equation but contains the information about all the higher order corrections, could serve as a model for the physical system, when some conditions on the parameters are imposed. It might provide an information about the influence of higher order corrections upon the leading order (soliton) dynamics.