Partial difference equations can also form integrable systems. Prime examples of such systems are integrable lattice equations, which are the difference analogues of the soliton equations. These cannot be obtained by any "brute force" discretisation method, as the latter will typically destroy the key integrability properties (such as the existence of soliton solutions). To obtain genuine integrable discrete systems that preserve the key properties much subtler methods are needed. The resulting equations turn out to be relevant to a wide range of applications in mathematical physics, numerical analysis and computer science, as well as mathematical biology, and economics.
The theory of difference equations has historically been less well developed than the analogous theories for differential equations. The development of rigorous analytic tools for difference equations is still in its infancy. It is here that the study of the integrability of discrete systems can provide new important insights, e.g. by developing a theory of symmetries and symmetry reductions for difference equations and by studying their singularity structures, as well as dealing with the problem of their classification. At Leeds we are investigating similarity reductions of partial difference equations, their initial value problems, as well as discrete analogues of the famous Painlevé transcendental equations. We are also studying discrete-time many-body systems, integrable dynamical mappings and their connection to special functions both on the classical as well as quantum level.
Hamiltonian theory is an important element of integrable systems, whether discrete, ordinary differential or partial differential equations. Important new examples of completely integrable classical mechanical systems were discovered by using "soliton techniques", such as Lax pairs and bi-Hamiltonian representations. However, a whole new theory of Poisson brackets for PDEs was developed, starting with the discovery of two Poisson bracket representations of the KdV equation. This is called the bi-Hamiltonian property and has been established for a large number of systems. It has become one of the signatures of integrability. Over the years large families of systems with two or more compatible Poisson brackets (known as multi-Hamiltonian in general) have been discovered.
To understand this proliferation of results more clearly, it is necessary to classify compatible Poisson brackets and their hierarchies of equations. One approach is to link the structures of the Poisson brackets with that of the corresponding Lax or zero curvature representation. This is quite systematic for "generic" systems, but there are many open problem related to ``reductions" (such as restrictions of a system to one of its invariant manifolds). In the case of first order hyperbolic systems (known as "systems of hydrodynamics type"), the Hamiltonian operators are first order differential operators, with geometric interpretation (related to pseudo-Riemannian manifolds). The classification problem here is related to the geometric properties of the corresponding surfaces/manifolds.
There are many cases of apparently disparate systems being connected through some transformation or through reduction/embedding to a lower/higher dimensional system. Any classification must take account of these relations if we are to avoid "double-counting". The relationships are often surprising, so of a great deal of interest in themselves. For instance, there is an embedding of a well known class of finite dimensional, completely integrable system, known as "Stäckel separable systems", to integrable systems of "hydrodynamics type". Adding electromagnetic terms into the Stäckel system produces new "non-homogeneous" systems of hydrodynamics type. The classification of such systems is a difficult problem as is an appropriate generalisation of the separation of variables for the finite dimensional system. Stäckel systems are examples of geodesic motion or the motion of a particle on some particular manifold, under the influence of some forces. Other geodesic equations, not necessarily separable, but with a high degree of symmetry, can be constructed through a Poisson bracket representation of Lie algebras.
For many differential equations a Bäcklund transformation may be constructed from canonical transformations. This enables us to establish connections between systems of integrable differential and difference equations. There are many starting points. We may use the Hamilton-Jacobi equation to construct a canonical transformation which preserves a given Hamiltonian function or seek canonical transformations or we may ask for transformations which preserve two or more compatible Poisson brackets.
The study of integrable quantum systems goes back to the beginnings of quantum mechanics. However, the quantum structure of discrete systems is a new development in this field. This work is being pursued in two directions: the quantum field theory of discrete systems on the space-time lattice, and the investigation of quantum Bäcklund transformations. The latter involves the study of special canonical transforms and their quantum analogues and provides a powerful tool for solving finite-dimensional quantum integrable systems.
In the classical theory (Lie, Liouville, etc.) of ordinary differential equations there are remarkable results which relate the property of integrability of ODEs in quadratures with the existence of continuous symmetries and first integrals. Symmetries and first integrals may serve as a solid mathematical foundation for an algebraic theory of integrable equations. In the case of integrable partial differential equations (PDEs) such a theory does already exist and proved to be extremely efficient. The key property of integrable PDEs is the existence of infinite hierarchies of local infinitesimal symmetries generated by a recursion operator. Characteristic features of integrable Hamiltonian PDEs are multi-Hamiltonian structures and hierarchies of local conservation laws. Study of these structures enable us to formulate constructive and very efficient tests for integrability and even solve the classificatin problem for some classes of equatiopns. A generalisation suitable for multi-dimensianal equations is in progress.
Data is transmitted along optical fibres as a series of pulses. Due to their non-interacting property, the ideal optical fibre allow the propagation of exact multi-soliton pulses. In the real world, however there are complications such as dissipation and a need for amplifiers. We are developing a perturbation theory to account for these effects.
For applications it is very important to consider the problem of propagation of nonlinear electromagnetic pulses in optical fibres with periodically alternating dispersion (so called Dispersion Managed, or DM, fibres). If the mean dispersion over the period vanishes and the amplitude of the pulse is small, then, according the linear theory, the complete compensation of the chromatic spreading would be achieved. In this case the map over the period of the DM fibre would be the identical transformation and the pulses would completely restore their profiles. However, to provide a low-error transmission and increase the signal-noise ratio one should use optical pulses of relatively high amplitude, thus the nonlinear effects become important and have to be accounted in the theoretical description. The solution of the above problem is very challenging theoretically and important for many applications including industrial.Return to Research Page
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