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Partially Integrable PT-Symmetric Hierarchies of Some Canonical Nonlinear Partial Differential Equations

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Date Issued:
2013
Abstract/Description:
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(&)#232; 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(&)#232; expansion for the solution.For the PT-symmetric 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(&)#228;cklund 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 Lax Pair.The PT-symmetric Burgers' equation fails the Painlev(&)#232; 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(&)#232; Test and Invariant Painlev(&)#232; analysis to 2+1 dimensions are utilized, and BTs and special solutions are found for those cases that pass the Painlev(&)#232; Test.
Title: Partially Integrable PT-Symmetric Hierarchies of Some Canonical Nonlinear Partial Differential Equations.
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Name(s): Pecora, Keri, Author
Choudhury, Sudipto, Committee Chair
Schober, Constance, Committee Member
Rollins, David, Committee Member
Christodoulides, Demetrios, Committee Member
University of Central Florida, Degree Grantor
Type of Resource: text
Date Issued: 2013
Publisher: University of Central Florida
Language(s): English
Abstract/Description: 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(&)#232; 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(&)#232; expansion for the solution.For the PT-symmetric 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(&)#228;cklund 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 Lax Pair.The PT-symmetric Burgers' equation fails the Painlev(&)#232; 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(&)#232; Test and Invariant Painlev(&)#232; analysis to 2+1 dimensions are utilized, and BTs and special solutions are found for those cases that pass the Painlev(&)#232; Test.
Identifier: CFE0004736 (IID), ucf:49843 (fedora)
Note(s): 2013-05-01
Ph.D.
Sciences, Mathematics
Doctoral
This record was generated from author submitted information.
Subject(s): pt-symmetric -- painleve
Persistent Link to This Record: http://purl.flvc.org/ucf/fd/CFE0004736
Restrictions on Access: campus 2016-05-15
Host Institution: UCF

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