Browsing by Subject "Fokas method"

Now showing 1 - 4 of 4

Open Access
Dispersion estimates for the boundary integral operator associated with the fourth order Schrödinger equation posed on the half line
(Element d.o.o., 2021-09) 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.
Open Access
Existence of unattainable states for Schrödinger type flows on the half-line
(Oxford University Press, 2023-12-01) Özsarı, Türker; Kalimeris, Konstantinos
We prove that the solutions of the Schrödinger and biharmonic Schrödinger equations do not have the exact boundary controllability property on the half-line by showing that the associated adjoint models lack observability. We consider the framework of L2 boundary controls with data spaces H−1(R+) and H−2(R+) for the classical and biharmonic Schrödinger equations, respectively. The lack of controllability on the half-line contrasts with the corresponding dynamics on a finite interval for a similar regularity setting. Our proof is based on an argument that uses the sharp fractional time trace estimates for solutions of the adjoint models. We also make several remarks on the connection of controllability and temporal regularity of spatial traces.
Open Access
The interior-boundary Strichartz estimate for the Schrödinger equation on the half-line revisited
(TÜBİTAK, 2022-01-01) Köksal, Bilge; Özsarı, Türker
In recent papers, it was shown for the biharmonic Schrödinger equation and 2D Schrödinger equation that Fokas method-based formulas are capable of defining weak solutions of associated nonlinear initial boundary value problems (ibvps) below the Banach algebra threshold. In view of these results, we revisit the theory of interiorboundary Strichartz estimates for the Schrödinger equation posed on the right half line, considering both Dirichlet and Neumann cases. Finally, we apply these estimates to obtain low regularity solutions for the nonlinear Schrödinger equation (NLS) with Neumann boundary condition and a coupled system of NLS equations defined on the half line with Dirichlet/Neumann boundary conditions. © This work is licensed under a Creative Commons Attribution 4.0 International License.
Open Access
Local well-posedness of the higher-order nonlinear Schrodinger equation on the half-line: single-boundary condition case
(Wiley-Blackwell Publishing, Inc, 2023-09-11) Alkın, A.; Mantzavinos, D.; Özsarı, Türker
We establish local well-posedness in the sense of Hadamard for a certain third-order nonlinear Schrodinger equation with a multiterm linear part and a general power nonlinearity, known as higher-order nonlinear Schrodinger equation, formulated on the half-line {x > 0}. 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 x = 0. Our functional framework centers around fractional Sobolev spaces H-x(s)(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 Schrodinger 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.