The Department of Solar Energy and Environmental Physics,

J. Blaustein Inst. for Desert Research,

Ben-Gurion University,

Sede Boker Campus 84990,

ISRAEL

E-mail: georg@bgu.ac.il

**Extended symmetries of the perturbed evolution equations**

**Abstract:**

The concept of extended symmetries ("$a$-symmetries") is introduced
for evolution equations dependent on a parameter ($a$). Mainly, the evolution
equations arising as a result of an asymptotic perturbation expansion with
the leading order term given by an integrable nonlinear equation are meant
For such equations the $a$-symmetries, as distinct from standard symmetries,
map solutions of the evolution equation with higher order corrections to
solutions of the leading order equation. Among the direct applications
of the $a$-symmetries are explicit formulas for calculation of the approximate
(standard) symmetries of the perturbed equation from the symmetries of
the leading order equation. The $a$-symmetries can be also used for construction
of the perturbed evolution equations possessing the naturally defined closed-form
asymptotic solutions. Applications to the KdV equation with higher-order
corrections are considered.