Symbolic Symmetry Analysis and
Its Applications
at
the International Conference on
Applications of Computer Algebra
July 21-22, 2007, Oakland
University, Rochester,
MI, USA
Organizers
Nicoleta Bila (Fayetteville State University) and Irina Kogan (North Carolina State University)
Motivation
Phenomena observed in nature have often symmetry
properties. These symmetry properties are inherited by the equations that model
these phenomena and can be exploited to either obtain explicit solutions, or
important geometric information about the solution set. Known today as symmetry
analysis, the symmetry reduction techniques, pioneered by Sophus
Lie, links the theory of differential equations with differential and
algebraic geometry, symbolic computation, numerical analysis, and
many other research fields. 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
Barbara Abraham-Shrauner
-
Willard Miller
-
Robert Milson -
Maria Clara Nucci -
Greg Reid
-
David A.
Richter -
Thomas
Wolf -
Schedule
The schedule of the talks is available from here
Talks
Barbara Abraham-Shrauner (Washington University, USA)
Type II hidden symmetries of nonlinear
partial differential equation
Abstract
The appearance or disappearance of
useful Lie symmetries of partial differential equations (PDEs)
has recently been studied where a symmetry is used to reduce the number of
variables. There are differences with
the inheritance of symmetries of PDEs and the
inheritance of symmetries of ordinary differential equations. We report on some
recent results of Type II hidden symmetries of nonlinear PDEs. These are new symmetries not inherited from
the original
Willard Miller (University of
Minnesota, USA)
Second order superintegrable
systems and algebraic varieties
Abstract
A classical (or
quantum) superintegrable system is an integrable Hamiltonian system on an n-dimensional
Riemannian space with potential that admits 2n-1 functionally independent constants
of the motion polynomial in the momenta, the maximum
number possible. If the constants are quadratic the system is second order superintegrable. The Kepler-Coulomb
system is the best known example. Such systems have remarkable properties:
multi-integrability and multi-separability,
an algebra of higher order symmetries whose representation theory yields
spectral information about the Schrödinger operator, deep connections with
special functions and with Quasi-Exactly Solvable systems. I will survey the
structure results for systems in 2 and 3 dimensions and show that for real and
complex Euclidean spaces the classification problem reduces to finding points
on an algebraic variety. Joint work with Ernie Kalnins
and Jonathan Kress.
Robert Milson (Dalhousie University, Canada)
Invariant subspaces of quadratically non-linear operators
Abstract
I will devote my talk to some new
structure theorems which characterize the spaces of linear and non-linear
differential operators that preserve the vector space P_n
of univariate polynomials of a degree n or less. In particular, we explicitly describe all quadratically non-linear differential operators Q(u_0,u_1,...,u_n) (where u_i denote the ith derivative of u) that map P_n
to itself. The main application is non-standard reductions of non-linear
evolution PDEs.
The proof technique is based on generating functions Correctness is
established by means of a
Maria Clara Nucci (University of Perugia, Italy)
Jacobi last multiplier and Lie symmetries: a stronghold of Physics
Abstract
The search for Lie symmetries admitted by a
system of ordinary differential equations may resemble the search for the Holy
Grail. Researchers often develop ad-hoc techniques, especially if they search
conservations laws and prefer to neglect Noether's
theorem. If this is not enough, Lagrangians (aka Hamiltonians), fundamental quantities in Classical
Mechanics, are elusive and seem to be confined to special physical cases only.
We show how Jacobi last multiplier comes to the
rescue. Lie symmetries, conservation laws, and (many) Lagrangians
can be easily found if one knows one (or more) Jacobi
last multiplier. A Computer Algebra System capable of calculating Lie
symmetries, evaluating determinants of square matrices, and integrating simple
functions is needed. Of course, a forget-me-not essential tool is a thinking
human brain.
Greg Reid (University of
Western Ontario, Canada)
Symmetry
computation in approximate and analytic spaces
Abstract
I will begin by discussing some well
established symbolic approaches for symmetry analysis. The latest developments
have been in applying approximate computation to symmetry problems. New
directions will be discussed including symmetry computation in approximate and
analytic spaces.
David A. Richter (Western
Michigan University, USA)
Lie superalgebras
of matrix differential operators
Click here to
download the Maple file
Thomas Wolf (Brock University, Canada)
Computer algebra challenges from the search for 3-dimensional scalar discrete integrable equations
Abstract
A central object of
interest in the relatively young field of discrete differential geometry are
so-called `face' relations between weights associated to corners and edges of
an n-dimensional quadrilateral. If such a relation allows the consistent
assembly of a (n+1)-dimensional discrete net then it is called consistent and
is fundamental in that smooth limits lead to integrable
PDEs and Baecklund
transformations between them. The talk starts with an introduction to discrete
differential geometry and continues with a description of the computational
problems and of probabilistic methods that had to be introduced in order to
handle the otherwise astronomically large polynomial algebraic system.
Irina Kogan (North Carolina State University, USA)
Differential and variational calculus in invariant frames
Abstract
Many
interesting systems of differential equations and variational
problems arising in geometry and physics admit a group of symmetries. As it was
first recognized by S. Lie, these problems can be rewritten in terms of groups
invariant objects: differential invariants, invariant differential forms, and
invariant differential operators. It is desirable from both computational and
theoretical points of view to use a group-invariant basis of differential
operators and differential forms to perform further computations with symmetric
systems.
This provides
a motivation for performing differential and variational
calculus on a jet bundle relative to a non-standard moving frame. Complexity of
the structure equations for a non-standard coframe
and non-commutativity of differential operators
present, however, both theoretical and computational challenge. We present new
symbolic algorithms for computing prolongation of vector fields, integration by
parts, Euler-Lagrange and Helmholtz operators, and Noether correspondence relative to a non-standard basis of
differential operators and differential forms. This talk is based on joint work
with Peter Olver and Ian Anderson.
Nicoleta Bila (Fayetteville State University, USA)
Nonclassical equivalence transformations
for a parameter identification problem
Abstract
Nonclassical equivalence transformations represent a class of symmetry reductions that can be associated with certain parameter identification problems represented by partial differential equations. These Lie groups of transformations allow us to reduce the dimension of the studied model, to relate the direct and the inverse problems, and to find new analytical solutions. For a particular parameter identification problem arising in heat conduction, we show how these new symmetry reductions might be incorporated into the associated boundary conditions as well. Our MAPLE routine GENDEFNC, which uses the package DESOLV (authors Carminati and Vu), has been updated for this propose. The output of GENDEFNC is the nonlinear system of the determining equations of the nonclassical equivalence transformations. Joint work with Jitse Niesen.
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Last updated: July 30, 2007