Volume 9, Issue Supplement 2, September 2001

Foreword

Barbara ABRAHAM-SHRAUNER

Pages: 1 - 9

The reduction of nonlinear ordinary differential equations by a combination of first integrals and Lie group symmetries is investigated. The retention, loss or even gain in symmetries in the integration of a nonlinear ordinary differential equation to a first integral are studied for several examples....

K ANDRIOPOULOS, P G L LEACH

Pages: 10 - 23

The complete symmetry groups of systems of linear second order ordinary differential equations are considered in the context of the simple harmonic oscillator. One finds that in general the representation of the complete symmetry group is not unique and in the particular case of a four-dimensional system...

T CERQUETELLI, N CICCOLI, M C NUCCI

Pages: 24 - 35

Realizations of four dimensional Lie algebras as vector fields in the plane are explcitly constructed. Fourth order ordinary differential equations which admit such Lie symmetry algebras are derived. The route to their integration is described.

the Painlevé, the Psi Series

Pages: 36 - 48

The classical (ARS) algorithm used in the Painlevé test picks up only those functions analytic in the complex plane. We complement it with an iterative algorithm giving the leading order and the next terms in all cases. This algorithm works both for an ascending series (about a singularity at finite...

I K, F M MAHOMED, C WAFO SOH

Pages: 49 - 59

We obtain a basis of joint or proper differential invariants for the scalar linear hperbolic partial differential equation in two independent variables by the infinitesimal method. The joint invariants of the hyperbolic equation consist of combinations of the coefficients of the equation and their derivatives...

A H KARA, F M MAHOMED

Pages: 60 - 72

The classical generation theorem of conservation laws from known ones for a system of differential equations which uses the action of a canonical LieBäcklund generator is extended to include any LieBäcklund generator. Also, it is shown that the Lie algebra of LieBäcklund symmetries of a conserved...

P G L LEACH, S É BOUQUET

Pages: 73 - 91

Cheb-Terrab and Roche (J. Sym. Comp. 27 (1999), 501519) presented what they termed a systematic algorithm for the construction of integrating factors for second order ordinary differential equations. They showed that there were instances of odinary differential equations without Lie point symmetries...

D P MASON

Pages: 92 - 101

A group invariant solution for a steady two-dimensional jet is derived by considering a linear combination of the Lie point symmetries of Prandtl's boundary layer equations for the jet. Only two Lie point symmetries contribute to the solution and the ratio of the constants in the linear combination is...

E MOMONIAT

Pages: 102 - 109

The method of multiple scales is used to introduce a small-time scale into the nolinear diffusion equation modelling the spreading of a thin liquid drop under gravity. The Lie group method is used to analyse the resulting system. An approximate group invariant solution and an approximation to the waiting-time...

M C NUCCI, P G L LEACH

Pages: 110 - 121

The symmetry approach to the determination of Jacobi's last multiplier is inverted to provide a source of additional symmetries for the EulerPoinsot system. These addtional symmetries are nonlocal. They provide the symmetries for the representation of the complete symmetry group of the system.

Miloslav ZNOJIL

Pages: 122 - 133

The Feshbach-type reduction of the Hilbert space to the physically most relevant "model" subspace is suggested as a means of a formal unification of the standard quantum mechanics with its recently proposed PT symmetric modification. The resuting "effective" Hamiltonians Heff (E) are always Hermitian,...