Differential invariants, hidden and conditional symmetry
Abstract
UDC 517.958:512.86
We provide a review on the development of hidden symmetry concept in the field of partial differential equations including a series of results previously obtained by the author. We also adduce new examples of classes of equations having type II hidden symmetry, and explain the nature of known non-classical symmetry of some equations.
We suggest an algorithm for description of classes of equations having specified conditional or hidden symmetry and/or reducible to equations with smaller number of independent variables by using a specific ansatz. We consider reductions that exist due to Lie, conditional and type II hidden symmetry. We also discuss relations between the concepts of hidden and conditional symmetry. It is proved that the type II hidden symmetry, which is previously regarded to be a special type of non-Lie symmetry, arises from the non-trivial $Q$-conditional symmetry of reduced equations. This approach allows not only to find the hidden symmetry and new reductions of known equations, but also makes it possible to describe a general form of equations from the specified $Q$-conditional and type II hidden symmetry.
As an example, we describe the general classes of equations with hidden and conditional symmetry under rotations in the Lorentz and Euclid groups, for which the relevant hidden and conditional symmetry allows reduction to radial equations with a smaller number of independent variables.
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