Journal of Nonlinear Mathematical Physics

Volume 9, Issue Supplement 2, September 2001

Special Issue in Honour of PGL Leach on the Occasion of His 60th Birthday

Foreword

1. Hidden Symmetries, First Integralsvand Reduction of Order of Nonlinear Ordinary Differential Equations

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....

2. The Economy of Complete Symmetry Groups for Linear Higher Dimensional Systems

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...

3. Four Dimensional Lie Symmetry Algebras and Fourth Order Ordinary Differential Equations

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.

4. Singularity Analysis and a Function Unifying

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...

5. Basis of Joint Invariants for (1 + 1) Linear Hyperbolic Equations

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...

6. A Basis of Conservation Laws for Partial Differential Equations

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...

7. Symmetries and Integrating Factors

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...

8. Group Invariant Solution and Conservation Law for a Free Laminar Two-Dimensional Jet

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...

9. Approximate Waiting-Time for a Thin Liquid Drop Spreading under Gravity

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...

10. Jacobi's Last Multiplier and the Complete Symmetry Group of the Euler­Poinsot System

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.

11. Should PT Symmetric Quantum Mechanics Be Interpreted as Nonlinear?

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,...