42,068 research outputs found
Nonlinearity and disorder: Classification and stability of nonlinear impurity modes
We study the effects produced by competition of two physical mechanisms of
energy localization in inhomogeneous nonlinear systems. As an example, we
analyze spatially localized modes supported by a nonlinear impurity in the
generalized nonlinear Schr\"odinger equation and describe three types of
nonlinear impurity modes --- one- and two-hump symmetric localized modes and
asymmetric localized modes --- for both focusing and defocusing nonlinearity
and two different (attractive or repulsive) types of impurity. We obtain an
analytical stability criterion for the nonlinear localized modes and consider
the case of a power-law nonlinearity in detail. We discuss several scenarios of
the instability-induced dynamics of the nonlinear impurity modes, including the
mode decay or switching to a new stable state, and collapse at the impurity
site.Comment: 18 pages, 22 figure
Sampling from a system-theoretic viewpoint: Part I - Concepts and tools
This paper is first in a series of papers studying a system-theoretic approach to the problem of reconstructing an analog signal from its samples. The idea, borrowed from earlier treatments in the control literature, is to address the problem as a hybrid model-matching problem in which performance is measured by system norms. In this paper we present the paradigm and revise underlying technical tools, such as the lifting technique and some topics of the operator theory. This material facilitates a systematic and unified treatment of a wide range of sampling and reconstruction problems, recovering many hitherto considered different solutions and leading to new results. Some of these applications are discussed in the second part
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