2,477 research outputs found
Symmetries of nonlinear ordinary differential equations: the modified Emden equation as a case study
Lie symmetry analysis is one of the powerful tools to analyze nonlinear
ordinary differential equations. We review the effectiveness of this method in
terms of various symmetries. We present the method of deriving Lie point
symmetries, contact symmetries, hidden symmetries, nonlocal symmetries,
-symmetries, adjoint symmetries and telescopic vector fields of a
second-order ordinary differential equation. We also illustrate the algorithm
involved in each method by considering a nonlinear oscillator equation as an
example. The connections between (i) symmetries and integrating factors and
(ii) symmetries and integrals are also discussed and illustrated through the
same example. The interconnections between some of the above symmetries, that
is (i) Lie point symmetries and -symmetries and (ii) exponential
nonlocal symmetries and -symmetries are also discussed. The order
reduction procedure is invoked to derive the general solution of the
second-order equation.Comment: 31 pages, To appear in the proceedings of NMI workshop on nonlinear
integrable systems and their applications which was held at Centre for
Nonlinear Dynamics, Tiruchirappalli, Indi
Darboux transformations for a 6-point scheme
We introduce (binary) Darboux transformation for general differential
equation of the second order in two independent variables. We present a
discrete version of the transformation for a 6-point difference scheme. The
scheme is appropriate to solving a hyperbolic type initial-boundary value
problem. We discuss several reductions and specifications of the
transformations as well as construction of other Darboux covariant schemes by
means of existing ones. In particular we introduce a 10-point scheme which can
be regarded as the discretization of self-adjoint hyperbolic equation
Systems of Hess-Appel'rot Type and Zhukovskii Property
We start with a review of a class of systems with invariant relations, so
called {\it systems of Hess--Appel'rot type} that generalizes the classical
Hess--Appel'rot rigid body case. The systems of Hess-Appel'rot type carry an
interesting combination of both integrable and non-integrable properties.
Further, following integrable line, we study partial reductions and systems
having what we call the {\it Zhukovskii property}: these are Hamiltonian
systems with invariant relations, such that partially reduced systems are
completely integrable. We prove that the Zhukovskii property is a quite general
characteristic of systems of Hess-Appel'rote type. The partial reduction
neglects the most interesting and challenging part of the dynamics of the
systems of Hess-Appel'rot type - the non-integrable part, some analysis of
which may be seen as a reconstruction problem. We show that an integrable
system, the magnetic pendulum on the oriented Grassmannian has
natural interpretation within Zhukovskii property and it is equivalent to a
partial reduction of certain system of Hess-Appel'rot type. We perform a
classical and an algebro-geometric integration of the system, as an example of
an isoholomorphic system. The paper presents a lot of examples of systems of
Hess-Appel'rot type, giving an additional argument in favor of further study of
this class of systems.Comment: 42 page
Software to compute infinitesimal symmetries of exterior differenial systems, with applications
A description is given of a software package to compute symmetries of partial differential equations, using computer algebra. As an application, the computation of higher-order symmetries of the classical Boussinesq equation is given leading to the recursion operator for symmetries in a straightforward way. Nonlocal symmetries for the Federbush model are obtained yielding the linearization of the model
Symbolic Computation of Variational Symmetries in Optimal Control
We use a computer algebra system to compute, in an efficient way, optimal
control variational symmetries up to a gauge term. The symmetries are then used
to obtain families of Noether's first integrals, possibly in the presence of
nonconservative external forces. As an application, we obtain eight independent
first integrals for the sub-Riemannian nilpotent problem (2,3,5,8).Comment: Presented at the 4th Junior European Meeting on "Control and
Optimization", Bialystok Technical University, Bialystok, Poland, 11-14
September 2005. Accepted (24-Feb-2006) to Control & Cybernetic
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