Robert Milson -
Maria Clara Nucci -
Barbara Abraham-Shrauner (Washington University, USA)
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
Willard Miller (University of Minnesota, USA)
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)
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
The proof technique is based on generating functions Correctness is
established by means of a
Maria Clara Nucci (University of Perugia, Italy)
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
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)
Click here to download the Maple file
Thomas Wolf (Brock University, Canada)
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)
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 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.
Last updated: July 30, 2007