14,099 research outputs found
Internal Modes of Solitons and Near-Integrable Highly-Dispersive Nonlinear Systems
The transition from integrable to non-integrable highly-dispersive nonlinear
models is investigated. The sine-Gordon and -equations with the
additional fourth-order spatial and spatio-temporal derivatives, describing the
higher dispersion, and with the terms originated from nonlinear interactions
are studied. The exact static and moving topological kinks and soliton-complex
solutions are obtained for a special choice of the equation parameters in the
dispersive systems. The problem of spectra of linear excitations of the static
kinks is solved completely for the case of the regularized equations with the
spatio-temporal derivatives. The frequencies of the internal modes of the kink
oscillations are found explicitly for the regularized sine-Gordon and
-equations. The appearance of the first internal soliton mode is
believed to be a criterion of the transition between integrable and
non-integrable equations and it is considered as the sufficient condition for
the non-trivial (inelastic) interactions of solitons in the systems.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
A Discontinuous Galerkin Method for Ideal Two-Fluid Plasma Equations
A discontinuous Galerkin method for the ideal 5 moment two-fluid plasma
system is presented. The method uses a second or third order discontinuous
Galerkin spatial discretization and a third order TVD Runge-Kutta time stepping
scheme. The method is benchmarked against an analytic solution of a dispersive
electron acoustic square pulse as well as the two-fluid electromagnetic shock
and existing numerical solutions to the GEM challenge magnetic reconnection
problem. The algorithm can be generalized to arbitrary geometries and three
dimensions. An approach to maintaining small gauge errors based on error
propagation is suggested.Comment: 40 pages, 18 figures
Bridging the gap between the Jaynes-Cummings and Rabi models using an intermediate rotating wave approximation
We present a novel approach called the intermediate rotating wave
approximation (IRWA), which employs a time-averaging method to encapsulate the
dynamics of light-matter interaction from strong to ultrastrong coupling
regime. In contrast to the ordinary rotating wave approximation, this method
addresses the co-rotating and counter-rotating terms separately to trace their
physical consequences individually, and thus establishes the continuity between
the Jaynes-Cummings model and the quantum Rabi model. We investigate IRWA in
near resonance and large detuning cases. Our IRWA not only agrees well with
both models in their respective coupling strengths, but also offers a good
explanation for their differences
Simulation of ultrasonic imaging with linear arrays in causal absorptive media
Rigorous and efficient numerical methods are presented for simulation of acoustic propagation in a medium where the absorption is described by relaxation processes. It is shown how FFT-based algorithms can be used to simulate ultrasound images in pulse-echo mode. General expressions are obtained for the complex wavenumber in a relaxing medium. A fit to measurements in biological media shows the appropriateness of the model. The wavenumber is applied to three FFT-based extrapolation operators, which are implemented in a weak form to reduce spatial aliasing. The influence of the absorptive medium on the quality of images obtained with a linear array transducer is demonstrated. It is shown that, for moderately absorbing media, the absorption has a large influence on the images, whereas the dispersion has a negligible effect on the images.\ud
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Structured Sparsity: Discrete and Convex approaches
Compressive sensing (CS) exploits sparsity to recover sparse or compressible
signals from dimensionality reducing, non-adaptive sensing mechanisms. Sparsity
is also used to enhance interpretability in machine learning and statistics
applications: While the ambient dimension is vast in modern data analysis
problems, the relevant information therein typically resides in a much lower
dimensional space. However, many solutions proposed nowadays do not leverage
the true underlying structure. Recent results in CS extend the simple sparsity
idea to more sophisticated {\em structured} sparsity models, which describe the
interdependency between the nonzero components of a signal, allowing to
increase the interpretability of the results and lead to better recovery
performance. In order to better understand the impact of structured sparsity,
in this chapter we analyze the connections between the discrete models and
their convex relaxations, highlighting their relative advantages. We start with
the general group sparse model and then elaborate on two important special
cases: the dispersive and the hierarchical models. For each, we present the
models in their discrete nature, discuss how to solve the ensuing discrete
problems and then describe convex relaxations. We also consider more general
structures as defined by set functions and present their convex proxies.
Further, we discuss efficient optimization solutions for structured sparsity
problems and illustrate structured sparsity in action via three applications.Comment: 30 pages, 18 figure
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