Contemporary Mathematics

Volume 160, 1994

Hidden Symmetries of Differential Equations

B. ABRAHAM-SHRAUNER

AND

A. GUO

ABSTRACf. Hidden symmetries of differential equations are discussed. These

are of two types for ODEs:

type

I are the symmetries which present in a higher-

order ODE are lost when that ODE is reduced

to

a lower-order ODE and type ll are

the symmetries not present in the higher-order ODE which appear when that ODE

is reduced

to

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

type

I hidden symmetries more general first-

order ODEs are reported. Applications to Vlasov-Maxwell equations and the

Painleve-Ince equation are mentioned.

1. Introduction

The hidden symmetry analysis discussed here is part of a general attempt

to

obtain analytical solutions of nonlinear differential equations (NLDEs) that arise

in

engineering, physics, biological and chemical systems. The basic approach is

to

find variable transformations that simplify the NLDEs

to

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

to

quadratures but

simplification may

be

very helpful also.

An

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.

This

paper is in fmal form and no version of it will

be

submitted for publication elsewhere.

1

©

1994 American Mathematical Society

0271-4132/94 $1.00

+

$.25 per page

http://dx.doi.org/10.1090/conm/160/01560

Volume 160, 1994

Hidden Symmetries of Differential Equations

B. ABRAHAM-SHRAUNER

AND

A. GUO

ABSTRACf. Hidden symmetries of differential equations are discussed. These

are of two types for ODEs:

type

I are the symmetries which present in a higher-

order ODE are lost when that ODE is reduced

to

a lower-order ODE and type ll are

the symmetries not present in the higher-order ODE which appear when that ODE

is reduced

to

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

type

I hidden symmetries more general first-

order ODEs are reported. Applications to Vlasov-Maxwell equations and the

Painleve-Ince equation are mentioned.

1. Introduction

The hidden symmetry analysis discussed here is part of a general attempt

to

obtain analytical solutions of nonlinear differential equations (NLDEs) that arise

in

engineering, physics, biological and chemical systems. The basic approach is

to

find variable transformations that simplify the NLDEs

to

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

to

quadratures but

simplification may

be

very helpful also.

An

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.

This

paper is in fmal form and no version of it will

be

submitted for publication elsewhere.

1

©

1994 American Mathematical Society

0271-4132/94 $1.00

+

$.25 per page

http://dx.doi.org/10.1090/conm/160/01560