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Buidling Lax Integrable VariableCoefficient Generalizations to Integrable PDEs and Exact Solutions to Nonlinear PDEs
 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 variablecoefficient generalizations of various Laxintegrable NLPDE hierarchies recently introduced in the literature. It is demonstrated that the technique yields Lax or Sintegrable nonlinear partial differential equations (PDEs) with both time and spacedependent 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 EstabrookWahlquist 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 variablecoefficient generalizations to the nonlinear Schrodinger (NLS) equation, derivative NLS equation, PTsymmetric NLS, fifthorder 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 KPII equations as instructive examples before using each method to derive nontrivial solutions to a microstructure PDE and two generalized PochhammerChree equations.
Title:  Buidling Lax Integrable VariableCoefficient 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 variablecoefficient generalizations of various Laxintegrable NLPDE hierarchies recently introduced in the literature. It is demonstrated that the technique yields Lax or Sintegrable nonlinear partial differential equations (PDEs) with both time and spacedependent 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 EstabrookWahlquist 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 variablecoefficient generalizations to the nonlinear Schrodinger (NLS) equation, derivative NLS equation, PTsymmetric NLS, fifthorder 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 KPII equations as instructive examples before using each method to derive nontrivial solutions to a microstructure PDE and two generalized PochhammerChree equations.  
Identifier:  CFE0006173 (IID), ucf:51144 (fedora)  
Note(s): 
20160501 Ph.D. Sciences, Mathematics Doctoral This record was generated from author submitted information. 

Subject(s):  Spatiotemporally varying nlpde  LaxIntegrable  singular manifold method  exact solution  
Persistent Link to This Record:  http://purl.flvc.org/ucf/fd/CFE0006173  
Restrictions on Access:  public 20160515  
Host Institution:  UCF 