Master of Science in Mathematics, Youngstown State University, 2011, Department of Mathematics and Statistics

Across the country, school students learn that ocean waves cause water particles to form looping paths, traveling in circles which become smaller as you look deeper underwater. In this paper, we investigate the approximations which are used to make this claim. Furthermore, we investigate closer approximation techniques which show that these looping paths actually propogate forward with the wave's motion. Finally, we investigate the specific case of the soliton, which causes particles underneath to travel in a forward-moving arc, with no looping motion at all. With this background, we examine the recent work of A. Constantin and collaborators, specifically his conclusion that our results for the soliton hold for any solitary wave.

Committee: George T. Yates PhD (Advisor); Jozsi Jalics PhD (Committee Member); Steven L. Kent PhD (Committee Member) Subjects: Applied Mathematics; Fluid Dynamics; Mathematics; Physics

PhD, University of Cincinnati, 2007, Arts and Sciences : Mathematical Sciences

The equation of Korteweg and de Vries was derived as a model for propagation of surface water waves along the channel. This also approximates the models in nonlinear studies that includes and balance weak nonlinear and dispersive effects. In particular, the Korteweg-de Vries (KdV) equation is commonly accepted as a mathematical model for the unidirectional propagation of small-amplitude long waves in a nonlinear dispersive medium. Laboratory experiments have shown that when nonlinear, dispersive waves are forced periodically at one end of the channel, the signal eventually becomes temporally periodic at each spatial point. From dynamical system point of view the damped forced KdV equations have been studied by Ghidaglia and Sell and You. They have proved the existence of global attractor for the system. Zhang studied a damped forced KdV-equation posed on a finite domain with homogeneous Dirichlet boundary conditions and proved that if the external excitation is time periodic with the small amplitude, then the system admits a unique time periodic solution, which forms a inertial manifold for the infinite dimensional dynamical system. This work studies the Korteweg-de Vries equation posed on a bounded domain. It is shown that if the boundary forcing is time periodic with small amplitude then the corresponding solution of the system eventually becomes time-periodic. If the system is considered without the initial condition as an infinite dimensional dynamical system in the Hilbert space L2(0,1), it is also shown that for a given small amplitude periodic boundary forcing, the system admits a unique time periodic solution, that as a limit cycle, is locally exponentially stable.

Committee: Dr. Bingyu Zhang (Advisor) Subjects: Mathematics

PhD, University of Cincinnati, 2021, Arts and Sciences: Mathematical Sciences

The fifth order KdV equations model the plasma waves and other dispersive phenomena when the contribution of the third order KdV-type dispersion is small. In recent years, the well-posedness of the fifth order KdV equations, that is existence, uniqueness of the solution, and continuous dependence of the solution on datum(initial value or boundary values), has been intensively studied by mathematicians using the methods already developed for the KdV equation on the real line, periodic domain, half-line, and bounded domain.

Committee: Bingyu Zhang Ph.D. (Committee Chair); Donald French Ph.D. (Committee Member); Michael Goldberg Ph.D. (Committee Member); Andrew Lorent Ph.D. (Committee Member) Subjects: Mathematics

PhD, University of Cincinnati, 2009, Arts and Sciences : Mathematical Sciences

The Korteweg-de Vries equation models unidirectional propagation of small finite amplitude long waves in a non-dispersive medium. The well-posedness, that is the existence, uniqueness of the solution, and continuous dependence on data, has been studied on unbounded,periodic, and bounded domains.This research focuses on an initial and boundary value problem (IBVP) for the Korteweg-de Vries (KdV) equation posed on a bounded interval with general nonhomogeneous boundary conditions. Using Kato smoothing properties of an associated linear problem and the contraction mapping principle, the IBVP is shown to be locally well-posed given several conditions on the parameters for the boundary conditions, in the L²-based Sobolev space Hs(0, 1) for any s≥0.

Committee: Bingyu Zhang PhD (Committee Chair); H Dumas PhD (Committee Member); Anthony Leung PhD (Committee Member); Philip Korman PhD (Committee Member) Subjects: Mathematics

Doctor of Philosophy, The Ohio State University, 2010, Mathematics

The standard mathematical model for the motion of surface waves in shallow water is the Euler equations for inviscid, incompressible flow, supplemented by free surface conditions. Without known explicit solutions in a simple form, simplifying assumptions are invoked to derive weakly nonlinear models and other variants that describe wave propagation. These models have provided good results in various applications, but their region of validity is not known precisely. The goal in this thesis is to assess the asymptotic errors in the KdV model by computing solutions to Euler's equations numerically and comparing them with the predictions from the KdV equation directly.

Committee: Gregory Baker PhD (Advisor); Saleh Tanveer PhD (Committee Member); Ching-Shan Chou PhD (Committee Member) Subjects: Mathematics