 00301 G. Cicogna , G. Gaeta
 Partial Liepoint symmetries of differential equations
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Jul 22, 00

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Abstract. When we consider a differential equation $\De=0$ whose set of
solutions is $\S$, a Liepoint exact symmetry of this is a Liepoint
invertible transformation $T$ such that $T(\S)=\S$, i.e. such that any
solution to $\De=0$ is tranformed into a (generally, different) solution to
the same equation; here we define {\it partial} symmetries of $\De=0$ as
Liepoint invertible transformations $T$ such that there
is a nonempty subset $\cR \subset \S$ such that $T(\cR) = \cR$, i.e.
such that there is a subset of solutions to $\De=0$ which are transformed
one into the other. We discuss how to determine both partial symmetries
and the invariant set $\cR \subset \S$, and show that our procedure is effective
by means of concrete examples. We also discuss relations with conditional
symmetries, and how our discussion applies to the special case of dynamical
systems. Our discussion will focus on continuous Liepoint partial
symmetries, but our approach would also be suitable for more general
classes of transformations; in the appendix we will discuss the case of
discrete partial symmetries.
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