Keywords

Pt symmetric, painleve

Abstract

In this dissertation, we generalize the work of Bender and co-workers to derive new partially-integrable hierarchies of various PT -symmetric, nonlinear partial differential equations. The possible integrable members are identified employing the Painlev´e Test, a necessary but not sufficient integrability condition, and are indexed by the integer n, corresponding to the negative of the order of the dominant pole in the singular part of the Painlev´e expansion for the solution. For the PT -symmetric Korteweg-de Vries (KdV) equation, as with some other hierarchies, the first or n = 1 equation fails the test, the n = 2 member corresponds to the regular KdV equation, while the remainder form an entirely new, possibly integrable hierarchy. Integrability properties of the n = 3 and n = 4 members, typical of partially-integrable systems, including B¨acklund Transformations, a ’near-Lax Pair’, and analytic solutions are derived. The solutions, or solitary waves, prove to be algebraic in form, and the extended homogeneous balance technique appears to be the most efficient in exposing the near-Lax Pair. The PT -symmetric Burgers’ equation fails the Painlev´e Test for its n = 2 case, but special solutions are nonetheless obtained. Also, PT -Symmetric hierarchies of 2+1 Burgers’ and Kadomtsev-Petviashvili equations, which may prove useful in applications are analyzed. Extensions of the Painlev´e Test and Invariant Painlev´e analysis to 2+1 dimensions are utilized, and BTs and special solutions are found for those cases that pass the Painlev´e Test.

Notes

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Graduation Date

2013

Semester

Spring

Advisor

Choudhury, S. Roy

Degree

Doctor of Philosophy (Ph.D.)

College

College of Sciences

Department

Mathematics

Degree Program

Mathematics

Format

application/pdf

Identifier

CFE0004736

URL

http://purl.fcla.edu/fcla/etd/CFE0004736

Language

English

Release Date

May 2016

Length of Campus-only Access

3 years

Access Status

Doctoral Dissertation (Open Access)

Subjects

Dissertations, Academic -- Sciences, Sciences -- Dissertations, Academic

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Included in

Mathematics Commons

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