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....
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...
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 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...
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...
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...
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...
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...
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...
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.
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,...