Ben-Gurion University, Jacob Blaustein Institute for Desert Research,

Sede-Boker Campus, 84990, ISRAEL

E-mail: georg@bgumail.bgu.ac.il

**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.