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On higher order extensions for the fractional Laplacian
The technique of Caffarelli and Silvestre, characterizing the fractional
Laplacian as the Dirichlet-to-Neumann map for a function U satisfying an
elliptic equation in the upper half space with one extra spatial dimension, is
shown to hold for general positive, non-integer orders of the fractional
Laplace operator, by showing an equivalence between the H^s norm on the
boundary and a suitable higher-order seminorm of U
Slow optical solitons via intersubband transitions in a semiconductor quantum well
We show the formation of bright and dark slow optical solitons based on
intersubband transitions in a semiconductor quantum well (SQW). Using the
coupled Schrodinger-Maxwell approach, we provide both analytical and numerical
results. Such a nonlinear optical process may be used for the control
technology of optical delay lines and optical buffers in the SQW solid-state
system. With appropriate parameters, we also show the generation of a large
cross-phase modulation (XPM). Since the the intersubband energy level can be
easily tuned by an external bias voltage, the present investigation may open
the possibility for electrically controlled phase modulator in the solid-state
system
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