Mathematics / Matematik

Permanent URI for this collectionhttps://hdl.handle.net/11147/8

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Author: search.filters.author.Alkan, Kıvılcım

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  • Article
    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.
  • Article
    Citation - WoS: 6
    Citation - Scopus: 7
    Integrable Systems From Inelastic Curve Flows in 2-And 3-Dimensional Minkowski Space
    (Taylor & Francis, 2016) Alkan, Kıvılcım; Anco, Stephen C.
    Integrable systems are derived from inelastic flows of timelike, spacelike, and null curves in 2-and 3- dimensional Minkowski space. The derivation uses a Lorentzian version of a geometrical moving frame method which is known to yield the modified Korteveg-de Vries (mKdV) equation and the nonlinear Schrodinger (NLS) equation in 2- and 3- dimensional Euclidean space, respectively. In 2-dimensional Minkowski space, time-like/space-like inelastic curve flows are shown to yield the defocusing mKdV equation and its bi-Hamiltonian integrability structure, while inelastic null curve flows are shown to give rise to Burgers' equation and its symmetry integrability structure. In 3-dimensional Minkowski space, the complex defocusing mKdV equation and the NLS equation along with their bi-Hamiltonian integrability structures are obtained from timelike inelastic curve flows, whereas spacelike inelastic curve flows yield an interesting variant of these two integrable equations in which complex numbers are replaced by hyperbolic (split-complex) numbers.