RISC,
Castle of Hagenberg,
Linz, Austria, July 27  30, 2008
Organizers
Motivation
Phenomena observed in nature often have symmetry properties. These
symmetry properties are inherited by the equations that model these phenomena
and can be exploited to either obtain explicit solutions, or an important
geometric information about the solution set. Symmetry analysis links various
research disciplines including differential equations, differential and
algebraic geometry, numerical analysis, and symbolic computation. Over the
years, new types of symmetries have been studied, such as nonclassical
symmetries, potential symmetries, and generalized symmetries, and used to
obtain new solutions of equations arising in mathematical physics, mathematical
biology, image processing, engineering, and financial mathematics. Many, but
far from all symmetry reduction techniques have been implemented, and many
theoretical and computational open problems remain. The aim of this special
session is to bring together researchers interested in symmetry analysis,
symbolic computation, and their applications.
Participants
Alexander Bihlo  University of
Vienna, Austria
Sergey Golovin  Lavrentyev
Institute of Hydrodynamics, Russia
Kesh Govinder  University of KwaZuluNatal, South Africa
Vladimir Kornyak  Joint Institute for
Nuclear Research, Russia
Michal Marvan  Silesian
University in Opava, Czech Republic
Sergey V.
Meleshko
 Suranaree
University of Technology, Thailand
Johannes Middeke  Research
Institute for Symbolic Computation, Austria
Anatoly Nikitin  Institute
of Mathematics of NAS, Ukraine
Juha Pohjanpelto  Oregon State
University, SUA
Roman Popovych  Institute of Mathematics of NAS, Ukraine & University of Vienna,
Austria
Ekaterina
Shemyakova  Research Institute for Symbolic Computation, Austria
Thomas Wolf  Brock University, Canada
Registration & Accommodation
Registration
http://www.risc.unilinz.ac.at/about/conferences/summer2008/registration/
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Conference
Schedule
Session Schedule
Talks
Alexander Bihlo
Symmetry methods in dynamic meteorology

The governing equations of dynamic meteorology are generally
nonlinear partial differential equations. They describe the evolution of
the atmospheric variables, including the velocity, vorticity and divergence
fields. Due to the high complexity of interaction of these fields the
overall dominating way to obtain solutions of these equations is numerical
integration. For conceptual understanding, however, as well as for
benchmark tests of numerical weather and climate prediction models, exact
solutions of the underlying equations are highly desirable. In this talk,
we aim to give examples of two methods to attack this problem, exemplified
with the inviscid barotropic vorticity equation (IBVE). The IBVE
qualitatively describes the behaviour of largescale, twodimensional fluid
dynamics of midlatitudes of the rotating earth. We will firstly present
groupinvariant solutions of the IBVE, which include often observed
planetary waves. Secondly, we will use induced symmetries of the Fourier
transformed version of the IBVE to obtain the minimal system of coupled
ordinary differential equations, derived by Edward Lorenz as a simple model
of interaction of a zonal flow with disturbances of this flow. This method
may be an example of a general way how to obtain consistent finitemode
approximations of partial differential equations.


Sergey
Golovin
Hierarchy of partially invariant solutions to differential equations

It is noticed, that the partially
invariant solution (PIS) of a system of differential equations in many
cases can be represented as an invariant reduction of some PIS of the
higher rank. This introduces a hierarchic structure in the set of all PISs
of a given system of differential equations. An equivalence of the twostep
and the direct ways of construction of PISs is proved. The hierarchy
simplifies the process of enumeration and symbolic analysis of partially
invariant submodels to the given system of differential equations. This
approach is demonstrated on examples of the hierarchies of regular PISs to
shallow water equations and ideal MHD equations.


Kesh Govinder (joint work
with Barbara AbrahamShrauner)
Hidden symmetries of PDEs

Type II hidden symmetries of partial differential equations (PDEs)
are point symmetries that arise unexpectedly when the number of variables
of PDEs are decreased. Here we show how to determine the origin of these
hidden symmetries as point symmetries of a master PDE. In addition, we show
that they arise from a number of different sources which include potential
and contact symmetries.


Vladimir
Kornyak
Gauge invariance in discrete dynamical
systems

The fundamental laws of physics are described from
the modern point of view by gauge theories with continuous gauge groups.
With some modification the gauge principle can (and should) be applied to
discrete dynamical systems with nontrivial symmetries also. We consider
specifics of introduction of gauge invariance into Wequivariant discrete
dynamical systems with finite sets of states , where W is the group of
symmetries of . Typical examples of such systems are cellular automata on
finite lattices… (see the pdf file
for the complete abstract)


Michal Marvan
On zero curvature
representations, the spectral parameter problem, and recursion operators

The
talk will discuss algorithmic aspects of classification of PDE possessing a
zero curvature representation with values in a given Lie algebra. In full
generality, the problem reduces to a quasilinear system of equations in
total derivatives.
Zero
curvature representations depending on a nonremovable parameter (spectral
parameter) are among the most important attributes of twodimensional
integrable systems. In many cases a given system comes with a parameterless
zero curvature representation. The problem whether a parameter can be
incorporated will be called the spectral parameter problem.
We shall show how to reduce the spectral parameter problem to a system of
linear equations in total derivatives. In case when the solution lies in
the same Lie algebra, we have a computable cohomological obstruction to
existence of a nonremovable parameter. On the other hand, even a
parameterless zero curvature representation usually allows us to find the
recursion operator, hence to insert the parameter.


Sergey V. Meleshko
Applications of Reduce
in equivalence problems of differential equations

Compatibility analysis is one of the main
tools of studying differential equations. One of the features of the
compatibility analysis is the extensive analytical manipulations involved
in the calculations. These manipulations consist of sequentially executing
such operations as prolongations of a system, substitution of complicated
expressions, and analysis of independency. The cumbersome part of these
calculations (or certainly part of it) can be entrusted to a computer. A
large body of literature exists on the topic of application of computer
algebra systems. The presentation is devoted to review my own experience of
applications of computer symbolic manipulations in problems related with
the compatibility analysis. In the presentation these applications are
roughly separated in the three parts: (a) total compatibility analysis; (b)
group analysis; (c) equivalence problems of differential equations. The
first two parts are shortly reviewed. New results obtained in the third
part are mostly considered in the presentation. The new results include the
following: (1) equivalence of secondorder ordinary differential equations
to canonical set of equations (equations admitting a Lie group or Painlevé
equations); (2) linearization problem of third and fourth order ordinary
differential equations; (3) equivalence of parabolic partial differential
equations to a canonical set of equations. These problems are characterized
by similar approach of solving. Their solutions are obtained by using
Reduce symbolic system of manipulations.


Johannes Middeke
A polynomial time
algorithm for computing the Jacobson form for matrices of differential
operators

We consider the
ring R = K[\partial; \id, \theta] of differential operators over a differential
field (K,\theta). It is wellknown that, if K has characteristic zero,
matrices over R can be transformed into Jacobson form
diag(1,...1,f,0,...,0). Yet the classical algorithms seem to be of
exponential time complexity. In this talk we present a new polynomialtime
algorithm using modular methods. Furthermore, using this algorithm we will achieve the form
diag(1,...,1,f,0,...,0) also for fields of nonzero characteristic.


Anatoly Nikitin
Symmetries in surface diffusion processes and modelling functions for
coverage profiles

An approach to theory of diffusion processes is proposed, which is
based on application of both the symmetry analysis and method of modelling
functions. An algorithm for construction of the modelling functions for
coverage profiles is suggested, which is based on the error functions
expansion (ERFE) of concentration curves. This approach is used to analysis
experimental results obtained for surface diffusion of Li absorbed by
Mo(112), but can be applied to many other diffusion systems.


Continuous pseudogroups appear as the infinite dimensional
counterparts of local Lie groups of transformations in various physical and
geometrical contexts, including gauge theories, Hamiltonian mechanics and
symplectic and Poisson geometry, conformal field theory, symmetry groups
partial differential equations, such as the NavierStokes and
KadomtsevPetviashvili equations of fluid mechanics and plasma physics,
image recognition, and geometric numerical integration. In this talk I will
report on my recent joint work with Peter Olver on extending the classical
moving frames method to infinite dimensional pseudogroups. As in the finite
dimensional case, moving frames can be employed to produce complete sets of
differential invariants for a pseudogroup action and to effectively analyze
the algebraic structure of such invariants. Moreover, I will discuss a
novel reduction method based on the moving frames algorithm for exterior
differential systems invariant under the action of a continuous pseudogroup
and describe its applications to constructing analytic solutions to systems
of partial differential equations.


The singular reduction operators, i.e., the singular
operators of nonclassical (conditional) symmetry, of partial differential
equations and all the possible reductions of these equations to firstorder
ordinary differential ones are exhaustively described. The separation of
singular families of reduction operators is shown to be an important step
of algorithmic construction of nonclassical symmetries with symbolic
calculation systems.


Ekaterina Shemyakova
Moving Frames for Laplace Invariants

The development of
symbolic methods for the factorization and integration of linear PDEs, many
of the methods being generalizations of the Laplace transformations method,
requires the finding of complete generating sets of invariants for the
corresponding linear operators and their systems with respect to the gauge
transformations. Within the theory of Laplacelike methods, there is no
uniform approach to this problem, though some individual invariants for
hyperbolic bivariate operators, and complete generating sets of invariants
for second and thirdorder hyperbolic bivariate ones have been obtained.
We show a systematic and much more efficient approach to the same problem
by application of movingframe methods. We give explicit formulae for
complete generating sets of invariants for second and thirdorder
bivariate linear operators, hyperbolic and nonhyperbolic, and also
demonstrate the approach for pairs of operators appearing in Darboux
transformations. Some applications to factorizations of Linear Partial
Differential Operators will be discussed also.


Thomas Wolf (joint work with
Sergey Tsarev)
Hyperdeterminants as
integrable 3D difference equations

Abstract
In this talk, coauthored with Sergey Tsarev, we give the basic
definitions and known theoretical results about hyperdeterminants, briefly
sketch their modern applications (quantum computing, biomathematics,
numerics, stochastic calculus etc.) and give a proof of integrability
(understood as 4dconsistency) of a difference equation defined by the
2x2x2 hyperdeterminant. Independently integrability of the 2x2x2
hyperdeterminant was stated by R.Kashaev(1996) and W.Schief (2003).
A natural conjecture is that all 2^n hyperdeterminants define
discrete integrable equations.
We prove that this conjecture already fails in the case of the
2x2x2x2 hyperdeterminant, computed recently by B.Sturmfels et. al
(www.arxiv.org, math.CO/0602149 v2 3 Oct 2006).


Nicoleta Bila
A special class of
symmetry reductions for PDEs involving arbitrary functions – PDF

Special symmetry reductions for PDEs
involving arbitrary functions depending only on the independent variables
are discussed. These group transformations can also be associated with
parameter identification problems described by PDEs. In fact, these
transformations called the extended classical symmetries of the model are
the classical Lie symmetries related to the equation in which both the
parameter and the data are regarded as dependent variables. The
relationship between these transformations and the weak equivalence
transformations (due to Torrisi and Traciná), the generalized equivalence
transformations (introduced by Meleshko), and the equivalence
transformations in the sense of Ovsiannikov is studied. For this type of
equations, any generalized equivalence transformation is an extended
classical symmetry and, excepting a particular case, the extended classical
symmetries differ from the weak equivalence transformations. The
determining equations of the generalized equivalence transformations can be
obtained from the determining equations of the extended classical
symmetries. Therefore, for these equations, any symbolic manipulation
program designed to find the classical Lie symmetries can also be used to
determine the generalized equivalence transformations.


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Hagenberg Map
Here is the link to the webpage for our session at ACA
2007
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Last updated: September 18,
2008