Is a nonclassical symmetry a symmetry?

Abstract

Various versions of the definition of nonclassical symmetries existing in the literature are analyzed. Comparing properties of Lie and nonclassical symmetries leads to the conclusion that in fact a nonclassical symmetry is not a symmetry in the usual sense. Hence the term "reduction operator" is suggested instead of the name "operator of nonclassical symmetries". It is shown that in contrast to the case of single partial differential equations a satisfactory definition of nonclassical symmetries for systems of such equations has not been proposed up to now. Moreover, the cardinality of essential nonclassical symmetries is discussed, taking into account equivalence relations on the entire set of nonclassical symmetries.

15 pages

References in corpus (2)

• Reduction Operators of Linear Second-Order Parabolic Equations
• Equivalence of Q-Conditional Symmetries under Group of Local Transformation

Cited by in corpus (6)

• Generalized conditional symmetries of evolution equations
• Conditional symmetries and exact solutions of nonlinear reaction-diffusion systems with non-constant diffusivities
• Classification of reduction operators and exact solutions of variable coefficient Newell-Whitehead-Segel equations
• Reduction operators of variable coefficient semilinear diffusion equations with an exponential source
• Group-theoretical analysis of variable coefficient nonlinear telegraph equations
• Symmetry Classification of quasi-linear PDE's. II: an exceptional case