4,655 research outputs found
Loop Groups and Discrete KdV Equations
A study is presented of fully discretized lattice equations associated with
the KdV hierarchy. Loop group methods give a systematic way of constructing
discretizations of the equations in the hierarchy. The lattice KdV system of
Nijhoff et al. arises from the lowest order discretization of the trivial,
lowest order equation in the hierarchy, b_t=b_x. Two new discretizations are
also given, the lowest order discretization of the first nontrivial equation in
the hierarchy, and a "second order" discretization of b_t=b_x. The former,
which is given the name "full lattice KdV" has the (potential) KdV equation as
a standard continuum limit. For each discretization a Backlund transformation
is given and soliton content analyzed. The full lattice KdV system has, like
KdV itself, solitons of all speeds, whereas both other discretizations studied
have a limited range of speeds, being discretizations of an equation with
solutions only of a fixed speed.Comment: LaTeX, 23 pages, 1 figur
Recent developments in rotary-balance testing of fighter aircraft configurations at NASA Ames Research Center
Two rotary balance apparatuses were developed for testing airplane models in a coning motion. A large scale apparatus, developed for use in the 12-Foot Pressure Wind tunnel primarily to permit testing at high Reynolds numbers, was recently used to investigate the aerodynamics of 0.05-scale model of the F-15 fighter aircraft. Effects of Reynolds number, spin rate parameter, model attitude, presence of a nose boom, and model/sting mounting angle were investigated. A smaller apparatus, which investigates the aerodynamics of bodies of revolution in a coning motion, was used in the 6-by-6 foot Supersonic Wind Tunnel to investigate the aerodynamic behavior of a simple representation of a modern fighter, the Standard Dynamic Model (SDM). Effects of spin rate parameter and model attitude were investigated. A description of the two rigs and a discussion of some of the results obtained in the respective test are presented
Complex trajectory method in time-dependent WKB
We present a significant improvement to a time-dependent WKB (TDWKB)
formulation developed by Boiron and Lombardi [JCP {\bf108}, 3431 (1998)] in
which the TDWKB equations are solved along classical trajectories that
propagate in the complex plane. Boiron and Lombardi showed that the method
gives very good agreement with the exact quantum mechanical result as long as
the wavefunction does not exhibit interference effects such as oscillations and
nodes. In this paper we show that this limitation can be overcome by
superposing the contributions of crossing trajectories. We also demonstrate
that the approximation improves when incorporating higher order terms in the
expansion. These improvements could make the TDWKB formulation a competitive
alternative to current time-dependent semiclassical methods
Mathematical modeling of the aerodynamic characteristics in flight dynamics
Basic concepts involved in the mathematical modeling of the aerodynamic response of an aircraft to arbitrary maneuvers are reviewed. The original formulation of an aerodynamic response in terms of nonlinear functionals is shown to be compatible with a derivation based on the use of nonlinear functional expansions. Extensions of the analysis through its natural connection with ideas from bifurcation theory are indicated
Quantifying the Effect of Non-Larmor Motion of Electrons on the Pressure Tensor
In space plasma, various effects of magnetic reconnection and turbulence
cause the electron motion to significantly deviate from their Larmor orbits.
Collectively these orbits affect the electron velocity distribution function
and lead to the appearance of the "non-gyrotropic" elements in the pressure
tensor. Quantification of this effect has important applications in space and
laboratory plasma, one of which is tracing the electron diffusion region (EDR)
of magnetic reconnection in space observations. Three different measures of
agyrotropy of pressure tensor have previously been proposed, namely,
, and . The multitude of contradictory measures has
caused confusion within the community. We revisit the problem by considering
the basic properties an agyrotropy measure should have. We show that
, and are all defined based on the sum of the
principle minors (i.e. the rotation invariant ) of the pressure tensor. We
discuss in detail the problems of -based measures and explain why they may
produce ambiguous and biased results. We introduce a new measure
constructed based on the determinant of the pressure tensor (i.e. the rotation
invariant ) which does not suffer from the problems of -based
measures. We compare with other measures in 2 and 3-dimension
particle-in-cell magnetic reconnection simulations, and show that can
effectively trace the EDR of reconnection in both Harris and force-free current
sheets. On the other hand, does not show prominent peaks in
the EDR and part of the separatrix in the force-free reconnection simulations,
demonstrating that does not measure all the non-gyrotropic
effects in this case, and is not suitable for studying magnetic reconnection in
more general situations other than Harris sheet reconnection.Comment: accepted by Phys. of Plasm
Consequences of Zeeman Degeneracy for van der Waals Blockade between Rydberg Atoms
We analyze the effects of Zeeman degeneracies on the long-range interactions
between like Rydberg atoms, with particular emphasis on applications to quantum
information processing using van der Waals blockade. We present a general
analysis of how degeneracies affect the primary error sources in blockade
experiments, emphasizing that blockade errors are sensitive primarily to the
weakest possible atom-atom interactions between the degenerate states, not the
mean interaction strength. We present explicit calculations of the van der
Waals potentials in the limit where the fine-structure interaction is large
compared to the atom-atom interactions. The results are presented for all
potential angular momentum channels invoving s, p, and d states. For most
channels there are one or more combinations of Zeeman levels that have
extremely small dipole-dipole interactions and are therefore poor candidates
for effective blockade experiments. Channels with promising properties are
identified and discussed. We also present numerical calculations of Rb and Cs
dipole matrix elements and relevant energy levels using quantum defect theory,
allowing for convenient quantitative estimates of the van der Waals
interactions to be made for principal quantum numbers up to 100. Finally, we
combine the blockade and van der Waals results to quantitatively analyze the
angular distribution of the blockade shift and its consequence for angular
momentum channels and geometries of particular interest for blockade
experiments with Rb.Comment: 16 figure
Four Symmetries of the KdV equation
We revisit the symmetry structure of integrable PDEs, looking at the specific
example of the KdV equation. We identify 4 nonlocal symmetries of KdV depending
on a parameter, which we call generating symmetries. We explain that since
these are nonlocal symmetries, their commutator algebra is not uniquely
determined, and we present three possibilities for the algebra. In the first
version, 3 of the 4 symmetries commute; this shows that it is possible to add
further (nonlocal) commuting flows to the standard KdV hierarchy. The second
version of the commutator algebra is consistent with Laurent expansions of the
symmetries, giving rise to an infinite dimensional algebra of hidden symmetries
of KdV. The third version is consistent with asymptotic expansions for large
values of the parameter, giving rise to the standard commuting symmetries of
KdV, the infinite hierarchy of "additional symmetries", and their traditionally
accepted commutator algebra (though this also suffers from some ambiguity as
the additional symmetries are nonlocal). We explain how the 3 symmetries that
commute in the first version of the algebra can all be regarded as
infinitesimal double B\"acklund transformations. The 4 generating symmetries
incorporate all known symmetries of the KdV equation, but also exhibit some
remarkable novel structure, arising from their nonlocality. We believe this
structure to be shared by other integrable PDEs.Comment: 22 page
Symmetry structure of integrable hyperbolic third order equations
We explore the application of generating symmetries, i.e. symmetries that
depend on a parameter, to integrable hyperbolic third order equations, and in
particular to consistent pairs of such equations as introduced by Adler and
Shabat (AS). Our main result is that different infinite hierarchies of
symmetries for these equations can arise from a single generating symmetry by
expansion about different values of the parameter. We illustrate this, and
study in depth the symmetry structure, for two examples. The first is an
equation related to the potential KdV equation taken from AS. The second is a
more general hyperbolic equation than the kind considered in AS. Both equations
depend on a parameter, and when this parameter vanishes they become part of a
consistent pair. When this happens, the nature of the expansions of the
generating symmetries needed to derive the hierarchies also changes.Comment: 21 page
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