More than a hundred years ago, the Norwegian mathematician Sophus Lie developed a symmetry-based approach to obtaining exact solutions of differential equations. Symmetry methods have great power and generality - indeed, nearly all well-known techniques for solving differential equations are special cases of Lie's methods. An introduction to symmetry-based solution techniques can be found in my book:
Symmetry Methods for Differential Equations: A Beginner's Guide.
A brief overview of the simplest symmetry methods can be found here.
The latter half of the 20th century saw a great renewal of interest in symmetry methods, as it was recognized that they are useful in a huge range of applications. Furthermore, new techniques based on symmetries are being developed by various research groups worldwide. Our work at Surrey has mainly been directed at discovering how to use (continuous) Lie symmetries to tackle discrete problems. Much of our research has focused on two areas: discrete symmetries and difference equations.
Many differential equations have discrete symmetries, which cannot be found by Lie's methods. Such symmetries are important in many applications, but (generally speaking) it is not possible to calculate them all directly from the symmetry condition. We have found that there is another approach that can be used if the differential equation has known Lie symmetries (a requirement that is satisfied by most differential equations that arise from mathematical models). Each discrete symmetry maps the set of Lie symmetries to itself; moreover the mapping is linear with constant coefficients. Consequently, matrix methods can be used to classify all possible mappings, which leads to a classification of all discrete symmetries. This idea was introduced in the following papers.
Hydon PE 1998
Discrete point symmetries of ordinary differential equations
Proc. Roy. Soc. Lond. A 454: 1961-1972
Hydon PE 2000
How to construct the discrete symmetries of partial differential equations
Eur. J. Appl. Math. 11: 515-527
An almost-complete classification of the discrete point symmetries of scalar ordinary differential equations can be found in the following paper; the results are also applicable to many scalar partial differential equations and systems of differential equations.
Laine-Pearson FE, Hydon PE 2003
Classification of matrices for discrete symmetries of ordinary differential equations
Stud. Appl. Math. 111:269-299
For further publications on discrete symmetries, please refer to my publications list.
At first sight, the idea of using continuous symmetries to obtain exact solutions of a difference equation seems nonsensical. If the independent variables are discrete, one cannot act (locally) on them with Lie symmetries. However, just as for differential equations, the dependent variables are continuous. Therefore a continuous symmetry group can act solely on the dependent variables; this idea was suggested by Maeda in 1987. To find the Lie symmetries, it is necessary to solve the linearized symmetry condition. For differential equations, the linearized symmetry condition is an overdetermined system of partial differential equations. Many computer algebra packages are available that will solve such systems. However the linearized symmetry condition for difference equations is a functional equation, which is not easy to split into an overdetermined system. Until recently, the only systematic approaches to solving this functional equation were based on series expansions. Some assumptions must be made about the validity of such expansions; moreover, they do not always yield all symmetries in a given class.
Work at Surrey has led to another strategy for solving the linearized symmetry condition, based on the technique of invariant differentiation. This method successively eliminates terms from the functional equation, leading to a system of differential equations that can be solved iteratively. Usually the calculations must be done with the aid of computer algebra. The output is a complete list of all Lie symmetries of a given class. For many well-known difference equations, the symmetries can be written in closed form. The following paper is an introduction to this technique.
Hydon PE 2000
Symmetries and first integrals of ordinary difference equations
Proc. Roy. Soc. Lond. A 456: 2835-2855
Conservation laws for partial differential equations can be constructed directly with the aid of an algebraic structure called the variational complex. This technique is more general than Noether's Theorem, because it does not rely on any special properties of the differential equation (such as a variational, Hamiltonian, or multisymplectic formulation). To develop a systematic method for obtaining conservation laws of difference equations, it was necessary to construct a variational complex based on difference operators. Prof E L Mansfield (Kent) and I have recently achieved this. Of the two papers below, the first is a detailed description of the new complex, whilst the second is an overview with an emphasis on applications.
Hydon PE, Mansfield EL 2004
A variational complex for difference equations
Found. Comp. Math. 4: 187-217
Mansfield EL, Hydon PE 2001
On a variational complex for difference equations
The Geometrical Study of Differential Equations, eds Joshua A. Leslie and Thierry P. Robart
American Mathematical Society, Providence, RI, 121-129
The discrete part of the variational complex is analogous to the de Rham complex. In the following preprint, we show that the tools of exterior calculus and cohomology have discrete analogues.
Mansfield EL, Hydon PE 2006
Difference forms
The differential variational complex and its difference counterpart are both locally exact; this is proved by constructing a homotopy operator. Such an operator makes it possible to construct conservation laws systematically provided that the adjoint of the linearized symmetry condition can be solved. For difference equations the technique of invariant differentiation enables all solutions of a given class to be found, subject only the limitations of the computer algebra system that is used to carry out the calculations. The easiest nontrivial applications are to difference equations with two independent variables, as described in the following papers.
Hydon PE 2001
Conservation laws of partial difference equations with two independent variables
J. Phys. A 34: 10347-10355
Rasin, OG, Hydon PE, 2005
Conservation laws of discrete KdV equation
Many partial differential equations have an underlying multisymplectic structure, which is a natural generalization of the symplectic structure for Hamiltonian systems of ordinary differential equations. The following paper describes an important relationship between the multisymplectic structure and a particular set of conservation laws. This relationship is used to define discrete multisymplectic systems, some of which can be used as structure-preserving numerical approximations to continuous systems.
Hydon PE 2005
Multisymplectic conservation laws for differential and differential-difference equations
Proc. Roy. Soc. Lond. A 461: 1627-1637
For further publications on difference equations, please refer to my publications list.