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Similarity Reductions and Integrable Lattice Equations

Walker, Alan James (2001) Similarity Reductions and Integrable Lattice Equations. PhD thesis, University of Leeds.

Walker_AJ_Applied_Mathematics_PhD_2001.pdf - Final eThesis - complete (pdf)

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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.

Item Type: Thesis (PhD)
Academic Units: The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds)
Identification Number/EthosID: uk.bl.ethos.628566
Depositing User: Digitisation Studio Leeds
Date Deposited: 28 Oct 2014 14:04
Last Modified: 25 Nov 2015 13:46
URI: http://etheses.whiterose.ac.uk/id/eprint/7190

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