### Abstract

In this thesis we construct solutions of the first three ultra-discrete Painlevé equations (uP_{I-3}, uP

_{II-1, }and uP

_{III}) from initial conditions. We use simple saw-tooth functions as initial solution intervals. We show that any piece-wise linear solution interval for uP

_{I-3}may be written in terms of simple saw-tooth functions. We then show that for a special case of solutions for uP

_{I-3 }generated from a compound initial solution interval Y(n), there exists a formula for the solution written in terms of the solutions generated from the component saw-tooth functions y

_{i}(n) into which Y(n) decomposes. We study the shapes produced by general solutions uP

_{II-1}generated from saw-tooth initial solution intervals and show a method to determine the remaining number of shapes produced as n → ∞ from the point n, thus showing that there is a finite amount of theses shapes produced. We also study a special case of uP

_{II-1}solutions dubbed "Staircase solutions" for the shape the solutions take when plotted. We construct a formula used to generate staircase type solutions of uP

_{II-1 }for α ε R. Finally we study the behaviour of general solutions of uP

_{III}by examining the iterates of its solutions as they intersect specific boundary points.Two appendices may be found towards the end of the thesis, the first of which contains various proofs which are not included in the main body of the thesis. The second Appendix contains several pieces of Mathematical code which my be used to generate the graphs displayed in each chapter.

Date of Award | Jan 2021 |
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Original language | English |

Supervisor | Thomas Kecker (Supervisor), Andrew Burbanks (Supervisor) & Galina Filipuk (Supervisor) |