Extended generalized equivalence groups of classes of linear ordinary differential equations

Abstract:
We recall equivalence properties (in terms of equivalence groupoids) of the class $\mathcal L$ of $r$th order ($r\geqslant2$) linear ordinary differential equations as well as of its subclasses $\mathcal L_1$, $\mathcal L_2$, $\mathcal A_1$ and $\mathcal A_2$ that are associated with the rational form, the Laguerre-Forsyth form, and the first and second Arnold canonical forms, respectively [1]. The corresponding classes of homogeneous equations were also studied from the point of view of admissible transformations in [1].

Since the entire class $\mathcal L$ and both the subclasses $\mathcal L_1$ and $\mathcal L_2$ are normalized with respect to their usual equivalence groups if $r\geqslant3$, this allowed us to classify Lie symmetries of $r$th order ($r\geqslant3$) linear ordinary differential equations using the algebraic method in three different ways.

The structure of the equivalence groupoids of the classes $\mathcal A_1$ and $\mathcal A_2$, where $r\geqslant3$, is more complicated since these classes are even not semi-normalized. This is why they are not usable for the group classification of the class $\mathcal L$ although these are the forms that are involved in the procedure of reducing the order of linear ordinary differential equations. Normalization properties of non-normalized classes of such equations are improved by their reparameterization related to fundamental sets of solutions. As a result, examples of generalized extended equivalence groups are constructed for the first time ever. At the same time, the reparameterization is not applicable to group classification of linear ordinary differential equations due to the complicated relation between old and new arbitrary elements, the complex involvement of arbitrary elements in the new representation of equations and the appearance of gauge equivalence transformations.

[1] Boyko V.M., Popovych R.O. and Shapoval N.M., Equivalence groupoids of classes of linear ordinary differential equations and their group classification, J. Phys. Conf. Ser. 621 (2015), 012001, 17 pp., for extended and revised version see arXiv:1403.6062.