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Buidling Lax Integrable Variable-Coefficient Generalizations to Integrable PDEs and Exact Solutions to Nonlinear PDEs

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Date Issued:
2016
Abstract/Description:
This dissertation is composed of two parts. In Part I a technique based on extended Lax Pairs isfirst considered to derive variable-coefficient generalizations of various Lax-integrable NLPDE hierarchies recently introduced in the literature. It is demonstrated that the technique yields Lax- or S-integrable nonlinear partial differential equations (PDEs) with both time- and space-dependent coefficients which are thus more general than almost all cases considered earlier via other methods such as the Painleve Test, Bell Polynomials, and various similarity methods. However, this technique, although operationally effective, has the significant disadvantage that, for any integrable system with spatiotemporally varying coefficients, one must 'guess' a generalization of the structure of the known Lax Pair for the corresponding system with constant coefficients. Motivated by the somewhat arbitrary nature of the above procedure, we present a generalization to the well known Estabrook-Wahlquist prolongation technique which provides a systematic procedure for the derivation of the Lax representation. In order to obtain a nontrivial Lax representation we must impose differential constraints on the variable coefficients present in the nlpde. The resulting constraints determine a class of equations which represent generalizations to a previously known integrable constant coefficient nlpde. We demonstrate the effectiveness of this technique by deriving variable-coefficient generalizations to the nonlinear Schrodinger (NLS) equation, derivative NLS equation, PT-symmetric NLS, fifth-order KdV, and three equations in the MKdV hierarchy. In Part II of this dissertation, we introduce three types of singular manifold methods which have been successfully used in the literature to derive exact solutions to many nonlinear PDEs extending over a wide range of applications. The singular manifold methods considered are: truncated Painleve analysis, Invariant Painleve analysis, and a generalized Hirota expansion method. We then consider the KdV and KP-II equations as instructive examples before using each method to derive nontrivial solutions to a microstructure PDE and two generalized Pochhammer-Chree equations.
Title: Buidling Lax Integrable Variable-Coefficient Generalizations to Integrable PDEs and Exact Solutions to Nonlinear PDEs.
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Name(s): Russo, Matthew, Author
Choudhury, Sudipto, Committee Chair
Moore, Brian, Committee Member
Schober, Constance, Committee Member
Christodoulides, Demetrios, Committee Member
University of Central Florida, Degree Grantor
Type of Resource: text
Date Issued: 2016
Publisher: University of Central Florida
Language(s): English
Abstract/Description: This dissertation is composed of two parts. In Part I a technique based on extended Lax Pairs isfirst considered to derive variable-coefficient generalizations of various Lax-integrable NLPDE hierarchies recently introduced in the literature. It is demonstrated that the technique yields Lax- or S-integrable nonlinear partial differential equations (PDEs) with both time- and space-dependent coefficients which are thus more general than almost all cases considered earlier via other methods such as the Painleve Test, Bell Polynomials, and various similarity methods. However, this technique, although operationally effective, has the significant disadvantage that, for any integrable system with spatiotemporally varying coefficients, one must 'guess' a generalization of the structure of the known Lax Pair for the corresponding system with constant coefficients. Motivated by the somewhat arbitrary nature of the above procedure, we present a generalization to the well known Estabrook-Wahlquist prolongation technique which provides a systematic procedure for the derivation of the Lax representation. In order to obtain a nontrivial Lax representation we must impose differential constraints on the variable coefficients present in the nlpde. The resulting constraints determine a class of equations which represent generalizations to a previously known integrable constant coefficient nlpde. We demonstrate the effectiveness of this technique by deriving variable-coefficient generalizations to the nonlinear Schrodinger (NLS) equation, derivative NLS equation, PT-symmetric NLS, fifth-order KdV, and three equations in the MKdV hierarchy. In Part II of this dissertation, we introduce three types of singular manifold methods which have been successfully used in the literature to derive exact solutions to many nonlinear PDEs extending over a wide range of applications. The singular manifold methods considered are: truncated Painleve analysis, Invariant Painleve analysis, and a generalized Hirota expansion method. We then consider the KdV and KP-II equations as instructive examples before using each method to derive nontrivial solutions to a microstructure PDE and two generalized Pochhammer-Chree equations.
Identifier: CFE0006173 (IID), ucf:51144 (fedora)
Note(s): 2016-05-01
Ph.D.
Sciences, Mathematics
Doctoral
This record was generated from author submitted information.
Subject(s): Spatiotemporally varying nlpde -- Lax-Integrable -- singular manifold method -- exact solution
Persistent Link to This Record: http://purl.flvc.org/ucf/fd/CFE0006173
Restrictions on Access: public 2016-05-15
Host Institution: UCF

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