Open AccessJournal Article
Nonlinear geometrical optics for oscillatory wave trains with a continuous oscillatory spectrum
TLDR
In this paper, the authors make the nonlinear geometrical optics for wave trains with such a continuous oscillatory spectrum, which requires the introduction of new spaces, which are Wiener algebras associated to spaces of vector-valued measures with bounded total variation.Abstract:
The frequency and the direction of propagation of an oscillatory wave train may be read on its oscillatory spectrum. Many works in geometrical optics allow the study of at most countable oscillatory spectra. In these works, the number of directions of propagation is therefore at most countable, while many physical effects would require a continuous infinity of directions of propagation. The goal of this paper is to make the nonlinear geometrical optics for wave trains with such a continuous oscillatory spectrum. This requires the introduction of new spaces, which are Wiener algebras associated to spaces of vector-valued measures with bounded total variation. We also make qualitative studies on the properties of wave trains with continuous oscillatory spectrum, and on the incidence of the nonlinearity on such oscillations. We finally suggest an application of the results of this paper to the study of both the spontaneous and the stimulated Raman scatterings.read more
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Book
Time averaging for the strongly confined nonlinear Schrödinger equation
TL;DR: In this article, the authors studied the limiting behavior of a nonlinear Schrodinger equation describing a 3-dimensional gas that is strongly confined along the vertical, z direction, and proved that the fast oscillations are almost-periodic in time, with values in a Sobolev-like space.
References
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Nonlinear optics
TL;DR: In this method, non-linear susceptibility tensors are introduced which relate the induced dipole moment to a power series expansion in field strengths and the various experimental observations are described and interpreted in terms of this formalism.
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On vector measures
Joe Diestel,B. Faires +1 more
TL;DR: In this paper, the Radon-Nikodym theorem is generalized to the case of strongly bounded vector measures, which is a generalization of a result due to E. Leonard and K. Sundaresan.
Journal ArticleDOI
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Weak Compactness and Vector Measures
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