GCRIS

Now showing 1 - 4 of 4

Citation - WoS: 11
Citation - Scopus: 8
Local Well-Posedness of the Higher-Order Nonlinear Schrödinger Equation on the Half-Line: Single-Boundary Condition Case
(Wiley, 2023) Alkın, Aykut; Mantzavinos, Dionyssios; Özsarı, Türker
We 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.
Citation - WoS: 3
Citation - Scopus: 3
Dispersion 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; Alkan, Kıvılcım; Kalimeris, Konstantinos
In 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.
Citation - WoS: 22
Citation - Scopus: 20
Fokas Method for Linear Boundary Value Problems Involving Mixed Spatial Derivatives
(Royal Society of Chemistry, 2020) Fokas, A. S.; Batal, Ahmet; Özsarı, Türker
We obtain solution representation formulae for some linear initial boundary value problems posed on the half space that involve mixed spatial derivative terms via the unified transform method (UTM), also known as the Fokas method. We first implement the method on the second-order parabolic PDEs; in this case one can alternatively eliminate the mixed derivatives by a linear change of variables. Then, we employ the method to biharmonic problems, where it is not possible to eliminate the cross term via a linear change of variables. A basic ingredient of the UTM is the use of certain invariant maps. It is shown here that these maps are well defined provided that certain analyticity issues are appropriately addressed.
Citation - WoS: 26
Citation - Scopus: 27
An Elementary Proof of the Lack of Null Controllability for the Heat Equation on the Half Line
(Elsevier, 2020) Kalimeris, Konstantinos; Özsarı, Türker
In this note, we give an elementary proof of the lack of null controllability for the heat equation on the half line by employing the machinery inherited by the unified transform, known also as the Fokas method. This approach also extends in a uniform way to higher dimensions and different initial-boundary value problems governed by the heat equation, suggesting a novel methodology for studying problems related to controllability. (C) 2020 Elsevier Ltd. All rights reserved.

Mathematics / Matematik

Browse

Filters

Settings

Sort By

Results per page

Search Results