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End date: 2019 × Start date: 2010 × DDC: 530 × Type: Article × Type: Text × WGL Institute: WIAS ×

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    Ultrashort optical pulse propagation in terms of analytic signal
    (New York, NY : Hindawi, 2011) Amiranashvili, Sh.; Demircan, A.
    We demonstrate that ultrashort optical pulses propagating in a nonlinear dispersive medium are naturally described through incorporation of analytic signal for the electric field. To this end a second-order nonlinear wave equation is first simplified using a unidirectional approximation. Then the analytic signal is introduced, and all nonresonant nonlinear terms are eliminated. The derived propagation equation accounts for arbitrary dispersion, resonant four-wave mixing processes, weak absorption, and arbitrary pulse duration. The model applies to the complex electric field and is independent of the slowly varying envelope approximation. Still the derived propagation equation posses universal structure of the generalized nonlinear Schrdinger equation (NSE). In particular, it can be solved numerically with only small changes of the standard split-step solver or more complicated spectral algorithms for NSE. We present exemplary numerical solutions describing supercontinuum generation with an ultrashort optical pulse.
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    Spectral Theory of Infinite Quantum Graphs
    (Cham (ZG) : Springer International Publishing AG, 2018) Exner, Pavel; Kostenko, Aleksey; Malamud, Mark; Neidhardt, Hagen
    We investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a close connection between spectral properties of a quantum graph and the corresponding properties of a certain weighted discrete Laplacian on the underlying discrete graph. Using this connection together with spectral theory of (unbounded) discrete Laplacians on infinite graphs, we prove a number of new results on spectral properties of quantum graphs. Namely, we prove several self-adjointness results including a Gaffney-type theorem. We investigate the problem of lower semiboundedness, prove several spectral estimates (bounds for the bottom of spectra and essential spectra of quantum graphs, CLR-type estimates) and study spectral types.