Volume 160, 1994
Hidden Symmetries of Differential Equations
ABSTRACf. Hidden symmetries of differential equations are discussed. These
are of two types for ODEs:
I are the symmetries which present in a higher-
order ODE are lost when that ODE is reduced
a lower-order ODE and type ll are
the symmetries not present in the higher-order ODE which appear when that ODE
a lower-order ODE. The inverse problem for hidden symmetries
is described for nonabelian two-parameter subgroups of the projective
group for type I hidden symmetries and for energy conserving equations for
type ll hidden symmetries. For the
I hidden symmetries more general first-
order ODEs are reported. Applications to Vlasov-Maxwell equations and the
Painleve-Ince equation are mentioned.
The hidden symmetry analysis discussed here is part of a general attempt
obtain analytical solutions of nonlinear differential equations (NLDEs) that arise
engineering, physics, biological and chemical systems. The basic approach is
find variable transformations that simplify the NLDEs
a form that has fewer
variables for partial differential equations (PDEs) or is of lower order for ordinary
differential equations (ODEs). Ideally one reduces the NLDEs
very helpful also.
alternate path is the reduction of the
nonlinear differential equation to a linear differential equation.
The variable transformations are determined by the symmetry analysis of the
NLDEs where the symmetry properties are given by Lie groups or the associated Lie
algebra. In the classical Lie method the point transformations of particular differential
equations are found; this is a direct method. Contact transformations or generalized
symmetries are also found by direct methods but are not considered here.
Hidden symmetries are symmetries of PDEs or ODEs not found by the Lie
classical method for Lie point groups or by the direct methods for contact or
generalized symmetries. This general definition includes previous results where in
most cases the term hidden symmetries was not used. The two examples that
1991 Mathematics Subject Classification. Primary 34A43, 22E70.
Supported in part by the Southwestern Bell Foundation Grant.
paper is in fmal form and no version of it will
submitted for publication elsewhere.
1994 American Mathematical Society
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