56 research outputs found
Lie Point Symmetries and Commuting Flows for Equations on Lattices
Different symmetry formalisms for difference equations on lattices are
reviewed and applied to perform symmetry reduction for both linear and
nonlinear partial difference equations. Both Lie point symmetries and
generalized symmetries are considered and applied to the discrete heat equation
and to the integrable discrete time Toda lattice
Lie point symmetries of differential--difference equations
We present an algorithm for determining the Lie point symmetries of
differential equations on fixed non transforming lattices, i.e. equations
involving both continuous and discrete independent variables. The symmetries of
a specific integrable discretization of the Krichever-Novikov equation, the
Toda lattice and Toda field theory are presented as examples of the general
method.Comment: 17 pages, 1 figur
Lie Symmetries and Exact Solutions of First Order Difference Schemes
We show that any first order ordinary differential equation with a known Lie
point symmetry group can be discretized into a difference scheme with the same
symmetry group. In general, the lattices are not regular ones, but must be
adapted to the symmetries considered. The invariant difference schemes can be
so chosen that their solutions coincide exactly with those of the original
differential equation.Comment: Minor changes and journal-re
Symmetries of the Continuous and Discrete Krichever-Novikov Equation
A symmetry classification is performed for a class of differential-difference equations depending on 9 parameters. A 6-parameter subclass of these equations is an integrable discretization of the Krichever-Novikov equation. The dimension n of the Lie point symmetry algebra satisfies 1≤n≤5. The highest dimensions, namely n=5 and n=4 occur only in the integrable cases
Contact transformations for difference schemes
We define a class of transformations of the dependent and independent
variables in an ordinary difference scheme. The transformations leave the
solution set of the system invariant and reduces to a group of contact
transformations in the continuous limit. We use a simple example to show that
the class is not empty and that such "contact transformations for discrete
systems" genuinely exist
Superintegrable Systems with a Third Order Integrals of Motion
Two-dimensional superintegrable systems with one third order and one lower
order integral of motion are reviewed. The fact that Hamiltonian systems with
higher order integrals of motion are not the same in classical and quantum
mechanics is stressed. New results on the use of classical and quantum third
order integrals are presented in Section 5 and 6.Comment: To appear in J. Phys A: Mathematical and Theoretical (SPE QTS5
Lie point symmetries of difference equations and lattices
A method is presented for finding the Lie point symmetry transformations
acting simultaneously on difference equations and lattices, while leaving the
solution set of the corresponding difference scheme invariant. The method is
applied to several examples. The found symmetry groups are used to obtain
particular solutions of differential-difference equations
Difference schemes with point symmetries and their numerical tests
Symmetry preserving difference schemes approximating second and third order
ordinary differential equations are presented. They have the same three or
four-dimensional symmetry groups as the original differential equations. The
new difference schemes are tested as numerical methods. The obtained numerical
solutions are shown to be much more accurate than those obtained by standard
methods without an increase in cost. For an example involving a solution with a
singularity in the integration region the symmetry preserving scheme, contrary
to standard ones, provides solutions valid beyond the singular point.Comment: 26 pages 7 figure
Solutions to the Optical Cascading Equations
Group theoretical methods are used to study the equations describing
\chi^{(2)}:\chi^{(2)} cascading. The equations are shown not to be integrable
by inverse scattering techniques. On the other hand, these equations do share
some of the nice properties of soliton equations. Large families of explicit
analytical solutions are obtained in terms of elliptic functions. In special
cases, these periodic solutions reduce to localized ones, i.e., solitary waves.
All previously known explicit solutions are recovered, and many additional ones
are obtainedComment: 21 page
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