Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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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, 2024) Alkın, A.; Mantzavinos, D.; Özsarı, T.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.Article Citation - WoS: 22Citation - Scopus: 20Fokas Method for Linear Boundary Value Problems Involving Mixed Spatial Derivatives(Royal Society of Chemistry, 2020) Fokas, A. S.; Batal, Ahmet; Özsarı, TürkerWe 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.Article Citation - WoS: 3Citation - Scopus: 3Blow-Up of Solutions of Nonlinear Schrödinger Equations With Oscillating Nonlinearities(American Institute of Mathematical Sciences, 2019) Özsarı, TürkerThe finite time blow-up of solutions for 1-D NLS with oscillating nonlinearities is shown in two domains: (1) the whole real line where the nonlinear source is acting in the interior of the domain and (2) the right half-line where the nonlinear source is placed at the boundary point. The distinctive feature of this work is that the initial energy is allowed to be non-negative and the momentum is allowed to be infinite in contrast to the previous literature on the blow-up of solutions with time dependent nonlinearities. The common finite momentum assumption is removed by using a compactly supported or rapidly decaying weight function in virial identities - an idea borrowed from [18]. At the end of the paper, a numerical example satisfying the theory is provided.Article Citation - WoS: 11Citation - Scopus: 10Pseudo-Backstepping and Its Application To the Control of Korteweg-De Vries Equation From the Right Endpoint on a Finite Domain(Society for Industrial and Applied Mathematics Publications, 2019) Özsarı, Türker; Batal, AhmetIn this paper, we design Dirichlet-Neumann boundary feedback controllers for the Korteweg-de Vries equation that act at the right endpoint of the domain. The length of the domain is allowed to be critical. Constructing backstepping controllers that act at the right endpoint of the domain is more challenging than its left endpoint counterpart. The standard application of the backstepping method fails, because corresponding kernel models become overdetermined. In order to deal with this difficulty, we introduce the pseudo-backstepping method, which uses a pseudo-kernel that satisfies all but one desirable boundary condition. Moreover, various norms of the pseudo-kernel can be controlled through a parameter in one of its boundary conditions. We prove that the boundary controllers constructed via this pseudo-kernel still exponentially stabilize the system with the cost of a low exponential rate of decay. We show that a single Dirichlet controller is sufficient for exponential stabilization with a slower rate of decay. We also consider a second order feedback law acting at the right Dirichlet boundary condition. We show that this approach works if the main equation includes only the third order term, while the same problem remains open if the main equation involves the first order and/or the nonlinear terms. At the end of the paper, we give numerical simulations to illustrate the main result.Article Citation - WoS: 40Citation - Scopus: 40The Initial-Boundary Value Problem for the Biharmonic Schrödinger Equation on the Half-Line(American Institute of Mathematical Sciences, 2019) Özsarı, Türker; Yolcu, NerminWe study the local and global wellposedness of the initial-boundary value problem for the biharmonic Schrodinger equation on the half-line with inhomogeneous Dirichlet-Neumann boundary data. First, we obtain a representation formula for the solution of the linear nonhomogenenous problem by using the Fokas method (also known as the unified transform method). We use this representation formula to prove space and time estimates on the solutions of the linear model in fractional Sobolev spaces by using Fourier analysis. Secondly, we consider the nonlinear model with a power type nonlinearity and prove the local wellposedness by means of a classical contraction argument. We obtain Strichartz estimates to treat the low regularity case by using the oscillatory integral theory directly on the representation formula provided by the Fokas method. Global wellposedness of the defocusing model is established up to cubic nonlinearities by using the multiplier technique and proving hidden trace regularities.Article Citation - WoS: 7Citation - Scopus: 7Output Feedback Stabilization of the Linearized Korteweg-De Vries Equation With Right Endpoint Controllers(Elsevier Ltd., 2019) Batal, Ahmet; Özsarı, TürkerIn this paper, we prove the output feedback stabilization for the linearized Korteweg-de Vries (KdV) equation posed on a finite domain in the case the full state of the system cannot be measured. We assume that there is a sensor at the left end point of the domain capable of measuring the first and second order boundary traces of the solution. This allows us to design a suitable observer system whose states can be used for constructing boundary feedbacks acting at the right endpoint so that both the observer and the original plant become exponentially stable. Stabilization of the original system is proved in the L-2-sense, while the convergence of the observer system to the original plant is also proved in higher order Sobolev norms. The standard backstepping approach used to construct a left endpoint controller fails and presents mathematical challenges when building right endpoint controllers due to the overdetermined nature of the related kernel models. In order to deal with this difficulty we use the method of Ozsan and Batal, (2019) which is based on using modified target systems involving extra trace terms. In addition, we show that the number of controllers and boundary measurements can be reduced to one, with the cost of a slightly lower exponential rate of decay. We provide numerical simulations illustrating the efficacy of our controllers. (C) 2019 Elsevier Ltd. All rights reserved.Article Citation - WoS: 6Citation - Scopus: 6Exponential Stability for the Nonlinear Schrodinger Equation With Locally Distributed Damping(Taylor and Francis Ltd., 2020) Cavalcanti, Marcelo M.; Correa, Wellington J.; Özsarı, Türker; Sepulveda, Mauricio; Vejar-Aseme, RodrigoIn this paper, we study the defocusing nonlinear Schrodinger equation with a locally distributed damping on a smooth bounded domain as well as on the whole space and on an exterior domain. We first construct approximate solutions using the theory of monotone operators. We show that approximate solutions decay exponentially fast in the L-2-sense by using the multiplier technique and a unique continuation property. Then, we prove the global existence as well as the L-2-decay of solutions for the original model by passing to the limit and using a weak lower semicontinuity argument, respectively. The distinctive feature of the paper is the monotonicity approach, which makes the analysis independent from the commonly used Strichartz estimates and allows us to work without artificial smoothing terms inserted into the main equation. We in addition implement a precise and efficient algorithm for studying the exponential decay established in the first part of the paper numerically. Our simulations illustrate the efficacy of the proposed control design.Article Citation - WoS: 26Citation - Scopus: 27An Elementary Proof of the Lack of Null Controllability for the Heat Equation on the Half Line(Elsevier, 2020) Kalimeris, Konstantinos; Özsarı, TürkerIn 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.Article Citation - WoS: 7Citation - Scopus: 7Complex Ginzburg–landau Equations With Dynamic Boundary Conditions(Elsevier Ltd., 2018) Corrêa, Wellington José; Özsarı, TürkerThe initial-dynamic boundary value problem (idbvp) for the complex Ginzburg–Landau equation (CGLE) on bounded domains of RN is studied by converting the given mathematical model into a Wentzell initial–boundary value problem (ibvp). First, the corresponding linear homogeneous idbvp is considered. Secondly, the forced linear idbvp with both interior and boundary forcings is studied. Then, the nonlinear idbvp with Lipschitz nonlinearity in the interior and monotone nonlinearity on the boundary is analyzed. The local well-posedness of the idbvp for the CGLE with power type nonlinearities is obtained via a contraction mapping argument. Global well-posedness for strong solutions is shown. Global existence and uniqueness of weak solutions are proven. Smoothing effect of the corresponding evolution operator is proved. This helps to get better well-posedness results than the known results on idbvp for nonlinear Schrödinger equations (NLS). An interesting result of this paper is proving that solutions of NLS subject to dynamic boundary conditions can be obtained as inviscid limits of the solutions of the CGLE subject to same type of boundary conditions. Finally, long time behavior of solutions is characterized and exponential decay rates are obtained at the energy level by using control theoretic tools.Article Citation - WoS: 7Citation - Scopus: 7Boosting the Decay of Solutions of the Linearised Korteweg-De Vries–burgers Equation To a Predetermined Rate From the Boundary(Taylor and Francis Ltd., 2019) Özsarı, Türker; Arabacı, EdaThe aim of this article is to extend recent results on the boundary feedback controllability of the Korteweg-de Vries equation to the Korteweg-de Vries–Burgers equation which is posed on a bounded domain. In the first part of the paper, it is proven that all the sufficiently small solutions can be steered to zero at any desired exponential rate by means of a suitably constructed boundary feedback controller. In the second part, an observer is proposed when a type of boundary measurement is available while there is no full access to the medium.