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Integrable delay-differential equations

Berntson, BK; (2017) Integrable delay-differential equations. Doctoral thesis, UCL (University College London). Green open access

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Abstract

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.

Type: Thesis (Doctoral)
Title: Integrable delay-differential equations
Event: University College London
Open access status: An open access version is available from UCL Discovery
Language: English
UCL classification: UCL > School of BEAMS
UCL > School of BEAMS > Faculty of Maths and Physical Sciences
URI: http://discovery.ucl.ac.uk/id/eprint/1566618
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