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Initial-Boundary Value Problem for the Higher-Order Nonlinear Schrödinger Equation on the Half-Line
(01. Izmir Institute of Technology, 2024) Alkın, Aykut; Batal, Ahmet; Özsarı, Türker
We establish local well-posedness in the sense of Hadamard for the higher-order nonlinear Schrödinger equation with a general power nonlinearity formulated on the halfline {x > 0}. We consider separately the two different scenarios of associated coefficients such that only one boundary condition is required, or exactly two boundary conditions are required. We assume a general nonhomogeneous boundary datum of Dirichlet type at x = 0 for the former case, and we add the Neumann type for the latter case. Our functional framework centers around fractional Sobolev spaces Hs x(R+) with respect to the spatial variable. We treat both high regularity (s > 1 2 ) and low regularity (s < 1 2 ) solutions: in the former setting, the relevant nonlinearity can be handled via the Banach algebra property; in the latter setting, however, this is no longer the case and, instead, delicate Strichartz estimates must be established. This task is especially challenging in the framework of nonhomogeneous initial-boundary value problems, as it involves proving boundary-type Strichartz estimates that are not common in the study of Cauchy (initial value) problems. The linear analysis, which forms the core of this work, crucially relies on a weak solution formulation defined through the novel solution formulae obtained via the Fokas method (also known as the unified transform) for the associated forced linear problem. In this connection, we note that the higher-order Schrödinger equation comes with an increased level of difficulty due to the presence of more than one spatial derivatives in the linear part of the equation. This feature manifests itself via several complications throughout the analysis, including (i) analyticity issues related to complex square roots, which require careful treatment of branch cuts and deformations of integration contours; (ii) singularities that emerge upon changes of variables in the Fourier analysis arguments; (iii) complicated oscillatory kernels in the weak solution formula for the linear initialboundary value problem, which require a subtle analysis of the dispersion in terms of the regularity of the boundary data.
Short Time Behaviour of Dam Break Flow Involving Two Liquids
(Izmir Institute of Technology, 2018) Isıdıcı Demirel, Damla; Yılmaz, Oğuz
The two dimensional dam break problem for wet bed case is investigated. The leading order and the second order problem are stated in nondimensional form. Solution to the leading order problem by using three different methods is given and explained in detail. Both Fourier series method and Galerkin method have difficulties on its own because of the singularity at the triple point. Although the singularity is ignored in Galerkin method, the method does not work except for the interface. Thus conformal mapping techniques is preferred because of the convenience and the strength of the complex analysis. The velocity profiles at whole boundary are obtained by using this conformal mapping. The second order solution of velocities are also obtained by using the same conformal mapping. On the other hand, the domain decomposition method (DDM) is applied for the second order dam break problem of dry bed case. The leading order solution helped to determine the suitable parameters for DDM. The leading order and second order solution of the free surfaces give a more realistic shape using the Lagrangian solution at the upper corner point. We assume this work contains useful and applicable methods in it for gravity driven flows and it will wake up different perspectives in readers mind.
Vortex Dynamics in Domains Whith Boundaries
(Izmir Institute of Technology, 2011) Tülü, Serdar; Yılmaz, Oğuz
In this thesis we consider the following problems: 1) The problem of fluid advection excited by point vortices in the presence of stationary cylinders (we also add a uniform flow to the systems). 2) The problem of motion of one vortex (or vortices) around cylinder(s). We also investigate integrable and chaotic cases of motion of two vortices around an oscillating cylinder in the presence of a uniform flow. In the fluid advection problems Milne-Thomson's Circle theorem and an analyticalnumerical solution in the form of an infinite power series are used to determine flow fields and the forces on the cylinder(s) are calculated by the Blasius theorem. In the "two vortices-one cylinder" case we generalize the problem by adding independent circulation k0 around the cylinder itself. We then write the conditions for force to be zero on the cylinder. The Hamiltonian for motion of two vortices in the case with no uniform flow and stationary cylinder is constructed and reduced. Also constant Hamiltonian (energy) curves are plotted when the system is shown to be integrable according to Liouville's definition. By adding uniform flow to the system and by allowing the cylinder to vibrate, we model the natural vibration of the cylinder in the flow field, which has applications in ocean engineering involving tethers or pipelines in a flow field. We conclude that in the chaotic case, forces on the cylinder may be considerably larger than those on the integrable case depending on the initial positions of the vortices, and that complex phenomena such as chaotic capture and escape occur when the initial positions lie in a certain region.

Phd Degree / Doktora

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