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.
| Identifer | oai:union.ndltd.org:ucf.edu/oai:stars.library.ucf.edu:etd-3909 |
| Date | 01 January 2013 |
| Creators | Pecora, Keri |
| Publisher | STARS |
| Source Sets | University of Central Florida |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Electronic Theses and Dissertations |
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