Nonlocal symmetries, telescopic vector fields and \(\lambda \)-symmetries of ordinary differential equations (Q352444)

This work analyzes the relation of standard and generalized \(\lambda\)-symmetries with other techniques to reduce systems of differential equations by means of the method of differential invariants and its generalizations, e.g. nonlocal symmetries, telescopic and exponential vector fields, etc. (see [\textit{P. J. Olver}, Applications of Lie groups to differential equations. New York, NY: Springer (2000; Zbl 0937.58026)] and [\textit{E. Pucci} and \textit{G. Saccomandi}, J. Phys. A, Math. Gen. 35, No. 29, 6145--6155 (2002; Zbl 1045.34014)]).NEWLINENEWLINEThe relation between telescopic vector fields and \(\lambda\)-symmetries is reviewed. Among other findings, it is shown that every telescopic vector field of a certain system of differential equations is related to a (generalized) \(\lambda\)-symmetry leaving invariant such an equation. Subsequently, the authors prove that a non-local symmetry obtained by the \(\lambda\)-covering method is associated to a \(\lambda\)-symmetry. In particular cases, such nonlocal symmetries are exponential vector fields, which can be related to \(\lambda\)-symmetries.NEWLINENEWLINEThe authors derive nonlocal symmetries of exponential type from \(\lambda\)-symmetries. Finally, they pioneer the investigation of the problem of determining when two reductions associated to two different nonlocal symmetries are strictly different. By using the notion of equivalence of \(\lambda\)-symmetries, the authors succeed in giving a simple criterion to solve this problem.NEWLINENEWLINEThese and other results are detailed and illustrated with interesting examples.