Similarity Reductions and Integrable Lattice Equations

In this thesis I extend the theory of integrable partial difference equations (PAEs)
and reductions of these systems under scaling symmetries. The main approach used is
the direct linearization method which was developed previously and forms a powerful
tool for dealing with both continuous and discrete equations. This approach is further
developed and applied to several important classes of integrable systems.
Whilst the theory of continuous integrable systems is well established, the theory of
analogous difference equations is much less advanced. In this context the study of
symmetry reductions of integrable (PAEs) which lead to ordinary difference equations
(OAEs) of Painleve type, forms a key aspect of a more general theory that is still in its
infancy.
The first part of the thesis lays down the general framework of the direct linearization
scheme and reviews previous results obtained by this method. Most results so far have
been obtained for lattice systems of KdV type. One novel result here is a new approach
for deriving Lax pairs. New results in this context start with the embedding of the
lattice KdV systems into a multi-dimensional lattice, the reduction of which leads
to both continuous and discrete Painleve hierarchies associated with the Painleve VI
equation.
The issue of multidimensional lattice equations also appears, albeit in a different way,
in the context of the lattice KP equations, which by dimensional reduction lead to new
classes of discrete equations.
This brings us in a natural way to a different class of continuous and discrete systems,
namely those which can be identified to be of Boussinesq (BSQ) type. The development
of this class by means of the direct linearization method forms one of the major parts of
the thesis. In particular, within this class we derive new differential-difference equations
and exhibit associated linear problems (Lax pairs). The consistency of initial value
problems on the multi-dimensional lattice is established. Furthermore, the similarity
constraints and their compatibility with the lattice systems guarantee the consistency
of the reductions that are considered. As such the resulting systems of lattice equations
are conjectured to be of Painleve type.
The final part of the thesis contains the general framework for lattice systems of AKNS
type for which we establish the basic equations as well as similarity constraints.