Scopus İndeksli Yayınlar Koleksiyonu / Scopus Indexed Publications Collection
Permanent URI for this collectionhttps://hdl.handle.net/11147/7148
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Article Citation - WoS: 11Citation - Scopus: 8Local Well-Posedness of the Higher-Order Nonlinear Schrödinger Equation on the Half-Line: Single-Boundary Condition Case(Wiley, 2023) Alkın, Aykut; Özsarı, Türker; Mantzavinos, Dionyssios; Özsarı, Türker; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyWe establish local well-posedness in the sense of Hadamard for a certain third-order nonlinear Schrödinger equation with a multiterm linear part and a general power nonlinearity, known as higher-order nonlinear Schrödinger equation, formulated on the half-line (Formula presented.). We consider the scenario of associated coefficients such that only one boundary condition is required and hence assume a general nonhomogeneous boundary datum of Dirichlet type at (Formula presented.). Our functional framework centers around fractional Sobolev spaces (Formula presented.) with respect to the spatial variable. We treat both high regularity ((Formula presented.)) and low regularity ((Formula presented.)) 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; and (iii) complicated oscillatory kernels in the weak solution formula for the linear initial-boundary value problem, which require a subtle analysis of the dispersion in terms of the regularity of the boundary data. The present work provides a first, complete treatment via the Fokas method of a nonhomogeneous initial-boundary value problem for a partial differential equation associated with a multiterm linear differential operator. © 2023 Wiley Periodicals LLC.Article Citation - WoS: 3Citation - Scopus: 3Dispersion Estimates for the Boundary Integral Operator Associated With the Fourth Order Schrödinger Equation Posed on the Half Line(Element d.o.o., 2022) Özsarı, Türker; Özsarı, Türker; Alkan, Kıvılcım; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyIn this paper, we prove dispersion estimates for the boundary integral operator associated with the fourth order Schr¨odinger equation posed on the half line. Proofs of such estimates for domains with boundaries are rare and generally require highly technical approaches, as opposed to our simple treatment which is based on constructing a boundary integral operator of oscillatory nature via the Fokas method. Our method is uniform and can be extended to other higher order partial differential equations where the main equation possibly involves more than one spatial derivatives.Article Citation - Scopus: 3Stabilization of Higher Order Schrödinger Equations on a Finite Interval: Part Ii(American Institute of Mathematical Sciences, 2022) Özsarı, Türker; Özsarı, Türker; Yılmaz, Kemal Cem; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyBackstepping based controller and observer models were designed for higher order linear and nonlinear Schrödinger equations on a finite interval in [3] where the controller was assumed to be acting from the left endpoint of the medium. In this companion paper, we further the analysis by considering boundary controller(s) acting at the right endpoint of the domain. It turns out that the problem is more challenging in this scenario as the associated boundary value problem for the backstepping kernel becomes overdetermined and lacks a smooth solution. The latter is essential to switch back and forth between the original plant and the so called target system. To overcome this difficulty we rely on the strategy of using an imperfect kernel, namely one of the boundary conditions in kernel PDE model is disregarded. The drawback is that one loses rapid stabilization in comparison with the left endpoint con-trollability. Nevertheless, the exponential decay of the L2-norm with a certain rate still holds. The observer design is associated with new challenges from the point of view of wellposedness and one has to prove smoothing properties for an associated initial boundary value problem with inhomogeneous boundary data. This problem is solved by using Laplace transform in time. However, the Bromwich integral that inverts the transformed solution is associated with certain analyticity issues which are treated through a subtle analysis. Numerical algorithms and simulations verifying the theoretical results are given.