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:
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. Much of my group's work has been directed at discovering how to use (continuous) Lie symmetries to tackle discrete problems. We have focused mainly on three areas: difference equations, multisymplectic systems and discrete symmetries.
It turns out that many of the basic solution techniques for differential equations can be transferred directly across to difference equations, though not usually in an obvious way. These include methods for finding and exploiting symmetries and conservation laws. The basic ideas are the same as for differential equations, but the methods have to be modified considerably. For a comprehensive introduction to this approach, see my latest book:
An introduction to methods for finding symmetries and conserved quantities in their simplest context can be found here.
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, Elizabeth Mansfield and I introduced the difference variational complex. The discrete part of the variational complex is analogous to the de Rham complex, with similar tools of difference exterior calculus and cohomology.
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, subject to 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; in particular, discrete integrable systems have an infinite hierarchy of conservation laws. Integrable partial differential equations are known to have infinite hierarchies of symmetries that are generated by a so-called mastersymmetry. The same is true for partial difference equations.
Where a given system of differential or difference is the Euler-Lagrange equation(s) for a variational problem, Noether's (First) Theorem links variational symmetries with conservation laws. For overdetermined differential systems with variational gauge symmetries, Noether's Second Theorem derives Bianchi-type identities from the gauge symmetries. Elizabeth Mansfield and I have discovered a difference analogue of Noether's Second Theorem, together with an intermediate result that bridges Noether's First and Second Theorems for both differential and difference systems.
For further publications on difference equations, please refer to my publications list.
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. An important relationship between the multisymplectic structure and a particular set of structural conservation laws can be used to define discrete multisymplectic systems, some of which can be used as structure-preserving numerical approximations to continuous systems. Just as symplectic systems have a conserved differential 2-form, multisymplectic systems have a conserved differential form, which is most naturally expressed in terms of the variational bicomplex. This leads to a coordinate-free formulation of multisymplecticity.
Working with Tom Bridges (who was among the first to define multisymplectic systems), Sebastian Reich, Darryl Holm and Colin Cotter, we have investigated applications of these ideas. In particular, we have discovered multisymplectic formulations of Lagrangian fluid models, the shallow water and semigeostrophic equations, and more general balanced meteorological models.
For further publications on multisymplectic systems, please refer to my publications list.
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. The mapping is associated with a Lie algebra automorphism that 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 works equally for ordinary and partial differential equations. Indeed, for scalar ordinary differential equations, Fiona Tomkinson (née Laine-Pearson) and I have created a classification of the discrete point symmetries that is almost complete.
Where a given differential equation has many independent Lie symmetries, classification of the Lie algebra automorphisms can be difficult. For this reason, David Fisher, Robert Gray and I have classified all automorphisms of Lie algebras of dimension 5 or less. The paper that includes this classification also establishes the Lie algebra version of the Krull-Schmidt Theorem, which explains how to all construct automorphisms of Lie algebras that can be written as direct sums of smaller components.
For further publications on discrete symmetries and Lie algebra automorphisms, please refer to my publications list.