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INTEGRABILITY OF A SINGULARLY PERTURBED MODEL DESCRIBING GRAVITY WATER WAVES ON A SURFACE OF FINITE DEPTH
 Date Issued:
 2008
 Abstract/Description:
 Our work is closely connected with the problem of splitting of separatrices (breaking of homoclinic orbits) in a singularly perturbed model describing gravity water waves on a surface of finite depth. The singularly perturbed model is a family of singularly perturbed fourthorder nonlinear ordinary differential equations, parametrized by an external parameter (in addition to the small parameter of the perturbations). It is known that in general separatrices will not survive a singular perturbation. However, it was proven by Tovbis and Pelinovsky that there is a discrete set of exceptional values of the external parameter for which separatrices do survive the perturbation. Since our family of equations can be written in the Hamiltonian form, the question is whether or not survival of separatrices implies integrability of the corresponding equation. The complete integrability of the system is examined from two viewpoints: 1) the existence of a second first integral in involution (Liouville integrability), and 2) the existence of singlevalued, meromorphic solutions (complex analytic integrability). In the latter case, a singular point analysis is done using the technique given by Ablowitz, Ramani, and Segur (the ARS algorithm) to determine whether the system is of Painlevétype (Ptype), lacking movable critical points. The system is shown by the algorithm to fail to be of Ptype, a strong indication of nonintegrability.
Title:  INTEGRABILITY OF A SINGULARLY PERTURBED MODEL DESCRIBING GRAVITY WATER WAVES ON A SURFACE OF FINITE DEPTH. 
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Name(s): 
Little, Steven, Author Tovbis, Alexander, Committee Chair University of Central Florida, Degree Grantor 

Type of Resource:  text  
Date Issued:  2008  
Publisher:  University of Central Florida  
Language(s):  English  
Abstract/Description:  Our work is closely connected with the problem of splitting of separatrices (breaking of homoclinic orbits) in a singularly perturbed model describing gravity water waves on a surface of finite depth. The singularly perturbed model is a family of singularly perturbed fourthorder nonlinear ordinary differential equations, parametrized by an external parameter (in addition to the small parameter of the perturbations). It is known that in general separatrices will not survive a singular perturbation. However, it was proven by Tovbis and Pelinovsky that there is a discrete set of exceptional values of the external parameter for which separatrices do survive the perturbation. Since our family of equations can be written in the Hamiltonian form, the question is whether or not survival of separatrices implies integrability of the corresponding equation. The complete integrability of the system is examined from two viewpoints: 1) the existence of a second first integral in involution (Liouville integrability), and 2) the existence of singlevalued, meromorphic solutions (complex analytic integrability). In the latter case, a singular point analysis is done using the technique given by Ablowitz, Ramani, and Segur (the ARS algorithm) to determine whether the system is of Painlevétype (Ptype), lacking movable critical points. The system is shown by the algorithm to fail to be of Ptype, a strong indication of nonintegrability.  
Identifier:  CFE0002109 (IID), ucf:47550 (fedora)  
Note(s): 
20080501 M.S. Sciences, Department of Mathematics Masters This record was generated from author submitted information. 

Subject(s): 
integrability Liouville integrability ARS algorithm Painleve property 

Persistent Link to This Record:  http://purl.flvc.org/ucf/fd/CFE0002109  
Restrictions on Access:  public  
Host Institution:  UCF 