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Integrable and non-integrable equations with peaked soliton solutions

Barnes, Lucy E. (2020) Integrable and non-integrable equations with peaked soliton solutions. Doctor of Philosophy (PhD) thesis, University of Kent,. (KAR id:79691)

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Abstract

This thesis explores a number of nonlinear PDEs that have peaked soliton solutions, to apply reductions to such PDEs and solve the resultant equations. Chapter 1 provides a brief history of peakon equations, where they come from and the different viewpoints of various authors. The rest of the chapter is then devoted to detailing the mathematical tools that will be used throughout the rest of the thesis. Chapter 2 concerns a coupling of two integrable peakon equations, namely the Popowicz system, which itself is not integrable. The 2-peakon dynamics are studied, and an explicit solution to the 2-peakon dynamics is given alongside some features of the interaction. In chapter 3 a reduction from two integrable peakon equations with quadratic nonlinearity to the third Painlev´e equation is given. B¨acklund transformations and solutions for the Painlev´e equations are expressed, and then used to find solutions of the original PDEs. A general peakon family, the b-family, is also explored, giving a more general result. Chapter 4 examines two peakon equations with cubic nonlinearity, and their reductions to Painlev´e equations. A link is shown between these cubic nonlinear peakon equations and the quadratic nonlinear equations in chapter 3. Chapter 5 has conclusions and outlook in the area.

Item Type: Thesis (Doctor of Philosophy (PhD))
Thesis advisor: Hone, Andrew
Uncontrolled keywords: Integrability Solitons Peakons Painleve equations
Divisions: Faculties > Sciences > School of Mathematics Statistics and Actuarial Science
SWORD Depositor: System Moodle
Depositing User: System Moodle
Date Deposited: 23 Jan 2020 11:10 UTC
Last Modified: 06 Feb 2020 04:20 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/79691 (The current URI for this page, for reference purposes)
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