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

Little, Steven, Tovbis, Alexander, University of Central Florida
 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...
Show moreOur 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.
Show less  Date Issued
 2008
 Identifier
 CFE0002109, ucf:47550
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0002109
 Title
 Spectral properties of the finite Hilbert transform on two adjacent intervals via the method of RiemannHilbert problem.
 Creator

Blackstone, Elliot, Tovbis, Alexander, Katsevich, Alexander, Tamasan, Alexandru, Pang, Sean, University of Central Florida
 Abstract / Description

In this dissertation, we study a selfadjoint integral operator $\hat{K}$ which is defined in terms of finite Hilbert transforms on two adjacent intervals. These types of transforms arise when one studies the interior problem of tomography. The operator $\hat{K}$ possesses a socalled ``integrable kernel'' and it is known that the spectral properties of $\hat{K}$ are intimately related to a $2\times2$ matrix function $\Gamma(z;\lambda)$ which is the solution to a particular RiemannHilbert...
Show moreIn this dissertation, we study a selfadjoint integral operator $\hat{K}$ which is defined in terms of finite Hilbert transforms on two adjacent intervals. These types of transforms arise when one studies the interior problem of tomography. The operator $\hat{K}$ possesses a socalled ``integrable kernel'' and it is known that the spectral properties of $\hat{K}$ are intimately related to a $2\times2$ matrix function $\Gamma(z;\lambda)$ which is the solution to a particular RiemannHilbert problem (in the $z$ plane). We express $\Gamma(z;\lambda)$ explicitly in terms of hypergeometric functions and find the small $\lambda$ asymptotics of $\Gamma(z;\lambda)$. This asymptotic analysis is necessary for the spectral analysis of the finite Hilbert transform on multiple adjacent intervals. We show that $\Gamma(z;\lambda)$ also has a jump in the $\lambda$ plane which allows us to compute the jump of the resolvent of $\hat{K}$. This jump is an important step in showing that the finite Hilbert transforms has simple and purely absolutely continuous spectrum. The well known spectral theory now allows us to construct unitary operators which diagonalize the finite Hilbert transforms. Lastly, we mention some future directions which include the many interval scenario and a bispectral property of $\hat{K}$.
Show less  Date Issued
 2019
 Identifier
 CFE0007602, ucf:52527
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0007602