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Title: Integrable delay-differential equations
Author: Berntson, B. K.
ISNI:       0000 0004 7225 4296
Awarding Body: UCL (University College London)
Current Institution: University College London (University of London)
Date of Award: 2017
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Delay-differential equations are differential-difference equations in which the derivatives and shifts are taken with respect to the same variable. This thesis is concerned with these equations from the perspective of the theory of integrable systems, and more specifically, Painlevé equations. Both the classical Painlevé equations and their discrete analogues can be obtained as deautonomizations of equations solved by two-parameter families of elliptic functions. In analogy with this paradigm, we consider autonomous delay-differential equations solved by elliptic functions, delay-differential extensions of the Painlevé equations, and the interrelations between these classes of equations. We develop a method to identify delay-differential equations that admit families of elliptic solutions with at least two degrees of parametric freedom and apply it to two natural 16-parameter families of delay-differential equations. Some of the resulting equations are related to known models including the differential-difference sine-Gordon equation and the Volterra lattice; the corresponding new solutions to these and other equations are constructed in a number of examples. Other equations we have identified appear to be new. Bäcklund transformations for the classical Painlevé equations provide a source of delay-differential Painlevé equations. These transformations were previously used to derive discrete Painlevé equations. We use similar methods to identify delay-differential equations with continuum limits to the first classical Painlevé equation. The equations we identify are solved by elliptic functions in particular limits corresponding to the autonomous limit of the classical first Painlevé equation.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available