On the Notion of Conditional Symmetry of Differential Equations

Abstract

Symmetry properties of PDE's are considered within a systematic and unifying scheme: particular attention is devoted to the notion of conditional symmetry, leading to the distinction and a precise characterization of the notions of "true" and "weak" conditional symmetry. Their relationship with exact and partial symmetries is also discussed. An extensive use of "symmetry-adapted" variables is made; several clarifying examples, including the case of Boussinesq equation, are also provided.

On the relation between standard and -symmetries for PDEs

“…In the case of partial µ-symmetries, instead, the PDE is transformed into an equation of the form (cf. [20,21,22], see also [2])…”

“…Then, the conditional symmetries for ∆ are simply those vector fields X for which (3.1) admits solutions. Note that X turns out to be a symmetry of the system (3.1) (see however [20,21,22] and also [2] for a careful discussion on the notion of conditional symmetries).…”

Symmetry Extensions and Their Physical Reasons in the Kinetic and Hydrodynamic Plasma Models

“…Strictly speaking, from the definition (6) it can be readily seen that the transformations generated by the operators (8), (9) in the space of variables t, x, v and f lead to the transformations generated by the infinitesimals (7) in the space of variables t, x and M k , k = 0, 1, 2, . .…”

“…The notion of conditional symmetry is discussed and large bibliography is presented in a recent paper [9].…”

Lambda and Mu-Symmetries

“…The concept of µ-symmetries is also generalized to an analogue of standard conditional and partial symmetries [3,1], i.e. partial (conditional) µ-symmetries [4].…”