3,237 research outputs found
"Dispersion management" for solitons in a Korteweg-de Vries system
The existence of ``dispersion-managed solitons'', i.e., stable pulsating
solitary-wave solutions to the nonlinear Schr\"{o}dinger equation with
periodically modulated and sign-variable dispersion is now well known in
nonlinear optics. Our purpose here is to investigate whether similar structures
exist for other well-known nonlinear wave models. Hence, here we consider as a
basic model the variable-coefficient Korteweg-de Vries equation; this has the
form of a Korteweg-de Vries equation with a periodically varying third-order
dispersion coefficient, that can take both positive and negative values. More
generally, this model may be extended to include fifth-order dispersion. Such
models may describe, for instance, periodically modulated waveguides for long
gravity-capillary waves. We develop an analytical approximation for solitary
waves in the weakly nonlinear case, from which it is possible to obtain a
reduction to a relatively simple integral equation, which is readily solved
numerically. Then, we describe some systematic direct simulations of the full
equation, which use the soliton shape produced by the integral equation as an
initial condition. These simulations reveal regions of stable and unstable
pulsating solitary waves in the corresponding parametric space. Finally, we
consider the effects of fifth-order dispersion.Comment: 19 pages, 7 figure
Coupled Ostrovsky equations for internal waves in a shear flow
In the context of fluid flows, the coupled Ostrovsky equations arise when two
distinct linear long wave modes have nearly coincident phase speeds in the
presence of background rotation. In this paper, nonlinear waves in a stratified
fluid in the presence of shear flow are investigated both analytically, using
techniques from asymptotic perturbation theory, and through numerical
simulations. The dispersion relation of the system, based on a three-layer
model of a stratified shear flow, reveals various dynamical behaviours,
including the existence of unsteady and steady envelope wave packets.Comment: 47 pages, 39 figures, accepted to Physics of Fluid
Wave Breaking and the Generation of Undular Bores in an Integrable Shallow Water System
The generation of an undular bore in the vicinity of a wave‐breaking point is considered for the integrable Kaup–Boussinesq (KB) shallow water system. In the framework of the Whitham modulation theory, an analytic solution of the Gurevich–Pitaevskii type of problem for a generic “cubic” breaking regime is obtained using a generalized hodograph transform, and a further reduction to a linear Euler–Poisson equation. The motion of the undular bore edges is investigated in detail
Evolution of solitary waves and undular bores in shallow-water flows over a gradual slope with bottom friction
This paper considers the propagation of shallow-water solitary and nonlinear periodic waves over a gradual slope with bottom friction in the framework of a variable-coefficient Korteweg-de Vries equation. We use the Whitham averaging method, using a recent development of this theory for perturbed integrable equations. This general approach enables us not only to improve known results on the adiabatic evolution of isolated solitary waves and periodic wave trains in the presence of variable topography and bottom friction, modelled by the Chezy law, but also, importantly, to study the effects of these factors on the propagation of undular bores, which are essentially unsteady in the system under consideration. In particular, it is shown that the combined action of variable topography and bottom friction generally imposes certain global restrictions on the undular bore propagation so that the evolution of the leading solitary wave can be substantially different from that of an isolated solitary wave with the same initial amplitude. This non-local effect is due to nonlinear wave interactions within the undular bore and can lead to an additional solitary wave amplitude growth, which cannot be predicted in the framework of the traditional adiabatic approach to the propagation of solitary waves in slowly varying media
Numerical study on diverging probability density function of flat-top solitons in an extended Korteweg-de Vries equation
We consider an extended Korteweg-de Vries (eKdV) equation, the usual
Korteweg-de Vries equation with inclusion of an additional cubic nonlinearity.
We investigate the statistical behaviour of flat-top solitary waves described
by an eKdV equation in the presence of weak dissipative disorder in the linear
growth/damping term. With the weak disorder in the system, the amplitude of
solitary wave randomly fluctuates during evolution. We demonstrate numerically
that the probability density function of a solitary wave parameter
which characterizes the soliton amplitude exhibits loglognormal divergence near
the maximum possible value.Comment: 8 pages, 4 figure
Robert Penn Warren in the 21st Century: The Good, the Bad, and the Ugly
Seven years into the 21st century, an informal look at the state of Warren studies reveals both reason for hope and for deep concern
Soliton formation from a pulse passing the zero-dispersion point in a nonlinear Schr\"odinger equation
We consider in detail the self-trapping of a soliton from a wave pulse that
passes from a defocussing region into a focussing one in a spatially
inhomogeneous nonlinear waveguide, described by a nonlinear Schrodinger
equation in which the dispersion coefficient changes its sign from normal to
anomalous. The model has direct applications to dispersion-decreasing nonlinear
optical fibers, and to natural waveguides for internal waves in the ocean. It
is found that, depending on the (conserved) energy and (nonconserved) mass of
the initial pulse, four qualitatively different outcomes of the pulse
transformation are possible: decay into radiation; self-trapping into a single
soliton; formation of a breather; and formation of a pair of counterpropagating
solitons. A corresponding chart is drawn on a parametric plane, which
demonstrates some unexpected features. In particular, it is found that any kind
of soliton(s) (including the breather and counterpropagating pair) eventually
decays into pure radiation with the increase of the energy, the initial mass
being kept constant. It is also noteworthy that a virtually direct transition
from a single soliton into a pair of symmetric counterpropagating ones seems
possible. An explanation for these features is proposed. In two cases when
analytical approximations apply, viz., a simple perturbation theory for broad
initial pulses, or the variational approximation for narrow ones, comparison
with the direct simulations shows reasonable agreement.Comment: 18 pages, 10 figures, 1 table. Phys. Rev. E, in pres
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Action, actor, context, target, time (AACTT): a framework for specifying behaviour
BACKGROUND: Designing implementation interventions to change the behaviour of healthcare providers and other professionals in the health system requires detailed specification of the behaviour(s) targeted for change to ensure alignment between intervention components and measured outcomes. Detailed behaviour specification can help to clarify evidence-practice gaps, clarify who needs to do what differently, identify modifiable barriers and enablers, design interventions to address these and ultimately provides an indicator of what to measure to evaluate an intervention's effect on behaviour change. An existing behaviour specification framework proposes four domains (Target, Action, Context, Time; TACT), but insufficiently clarifies who is performing the behaviour (i.e. the Actor). Specifying the Actor is especially important in healthcare settings characterised by multiple behaviours performed by multiple different people. We propose and describe an extension and re-ordering of TACT to enhance its utility to implementation intervention designers, practitioners and trialists: the Action, Actor, Context, Target, Time (AACTT) framework. We aim to demonstrate its application across key steps of implementation research and to provide tools for its use in practice to clarify the behaviours of stakeholders across multiple levels of the healthcare system. METHODS AND RESULTS: We used French et al.'s four-step implementation process model to describe the potential applications of the AACTT framework for (a) clarifying who needs to do what differently, (b) identifying barriers and enablers, (c) selecting fit-for-purpose intervention strategies and components and (d) evaluating implementation interventions. CONCLUSIONS: Describing and detailing behaviour using the AACTT framework may help to enhance measurement of theoretical constructs, inform development of topic guides and questionnaires, enhance the design of implementation interventions and clarify outcome measurement for evaluating implementation interventions
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