Symbolic Symmetry Analysis and Its Applications

at the International Conference on Applications of Computer Algebra

July 21-22, 2007, Oakland University, Rochester, MI, USA


Nicoleta Bila (Fayetteville State University) and Irina Kogan (North Carolina State University)


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.


Meadow Brook Hall Barbara Abraham-Shrauner - Washington University, USA


Willard Miller - University of Minnesota, USA


Robert Milson - Dalhousie University, Canada


Maria Clara Nucci -University of Perugia, Italy


Greg Reid - University of Western Ontario, Canada


David A. Richter - Western Michigan University, USA


Thomas Wolf - Brock University, Canada





The schedule of the talks is available from here


Barbara Abraham-Shrauner (Washington University, USA)

Type II hidden symmetries of nonlinear partial differential equation    



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 PDE when a symmetry is used to reduce the number of variables of the original PDE.  Examples include the two-dimensional  Burger’s equation, the four-dimensional second heavenly equation of gravitation and some model nonlinear equations.  The provenance of the Type II hidden symmetries has been shown for PDEs to be the inherited Lie point symmetries of PDEs other than the original PDE that reduce to the same target PDE.  Some results of the master equation for the other PDEs are given for a simple model nonlinear PDE.  Joint work with Keshlan S. Govinder.



Willard Miller (University of Minnesota, USA)

Second order superintegrable systems and algebraic varieties



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



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 CAS.



Maria Clara Nucci (University of Perugia, Italy)

Jacobi last multiplier and Lie symmetries: a stronghold of Physics   



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)

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



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



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



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