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INTEGRABILITY OF A SINGULARLY PERTURBED MODEL DESCRIBING GRAVITY WATER WAVES ON A SURFACE OF FINITE DEPTH

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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 fourth-order 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 single-valued, 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 Painlevé-type (P-type), lacking movable critical points. The system is shown by the algorithm to fail to be of P-type, 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 fourth-order 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 single-valued, 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 Painlevé-type (P-type), lacking movable critical points. The system is shown by the algorithm to fail to be of P-type, a strong indication of nonintegrability.
Identifier: CFE0002109 (IID), ucf:47550 (fedora)
Note(s): 2008-05-01
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

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