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.