100,504 research outputs found
Superposition rules for higher-order systems and their applications
Superposition rules form a class of functions that describe general solutions
of systems of first-order ordinary differential equations in terms of generic
families of particular solutions and certain constants. In this work we extend
this notion and other related ones to systems of higher-order differential
equations and analyse their properties. Several results concerning the
existence of various types of superposition rules for higher-order systems are
proved and illustrated with examples extracted from the physics and mathematics
literature. In particular, two new superposition rules for second- and
third-order Kummer--Schwarz equations are derived.Comment: (v2) 33 pages, some typos corrected, added some references and minor
commentarie
High-order rogue waves of a long wave-short wave model
The long wave-short wave model describes the interaction between the long
wave and the short wave. Exact higher-order rational solution expressed by
determinants is calculated via the Hirota's bilinear method and the KP
hierarchy reduction. It is found that the fundamental rogue wave for the short
wave can be classified into three different patterns: bright, intermediate and
dark ones, whereas the rogue wave for the long wave is always bright type. The
higher-order rogue waves correspond to the superposition of fundamental rogue
waves. The modulation instability analysis show that the condition of the
baseband modulation instability where an unstable continuous-wave background
corresponds to perturbations with infinitesimally small frequencies, coincides
with the condition for the existence of rogue-wave solutions.Comment: 14 pages, 5 figure
Superpositions of SU(3) coherent states via a nonlinear evolution
We show that a nonlinear Hamiltonian evolution can transform an SU(3)
coherent state into a superposition of distinct SU(3) coherent states, with a
superposition of two SU(2) coherent states presented as a special case. A phase
space representation is depicted by projecting the multi-dimensional -symbol
for the state to a spherical subdomain of the coset space. We discuss
realizations of this nonlinear evolution in the contexts of nonlinear optics
and Bose--Einstein condensates
Dirac--Lie systems and Schwarzian equations
A Lie system is a system of differential equations admitting a superposition
rule, i.e., a function describing its general solution in terms of any generic
set of particular solutions and some constants. Following ideas going back to
the Dirac's description of constrained systems, we introduce and analyse a
particular class of Lie systems on Dirac manifolds, called Dirac--Lie systems,
which are associated with `Dirac--Lie Hamiltonians'. Our results enable us to
investigate constants of the motion, superposition rules, and other general
properties of such systems in a more effective way. Several concepts of the
theory of Lie systems are adapted to this `Dirac setting' and new applications
of Dirac geometry in differential equations are presented. As an application,
we analyze traveling wave solutions of Schwarzian equations, but our methods
can be applied also to other classes of differential equations important for
Physics.Comment: 41 page
Photons in polychromatic rotating modes
We propose a quantum theory of rotating light beams and study some of its
properties. Such beams are polychromatic and have either a slowly rotating
polarization or a slowly rotating transverse mode pattern. We show there are,
for both cases, three different natural types of modes that qualify as
rotating, one of which is a new type not previously considered. We discuss
differences between these three types of rotating modes on the one hand and
non-rotating modes as viewed from a rotating frame of reference on the other.
We present various examples illustrating the possible use of rotating photons,
mostly for quantum information processing purposes. We introduce in this
context a rotating version of the two-photon singlet state.Comment: enormously expanded: 12 pages, 3 figures; a new, more informative,
but less elegant title, especially designed for Phys. Rev.
Fluctuations, Ghosts, and the Cosmological Constant
For a large region of parameter space involving the cosmological constant and
mass parameters, we discuss fluctuating spacetime solutions that are
effectively Minkowskian on large time and distance scales. Rapid, small
amplitude oscillations in the scale factor have a frequency determined by the
size of a negative cosmological constant. A field with modes of negative energy
is required. If it is gravity that induces a coupling between the ghost-like
and normal fields, we find that this results in stochastic rather than unstable
behavior. The negative energy modes may also permit the existence of Lorentz
invariant fluctuating solutions of finite energy density. Finally we consider
higher derivative gravity theories and find oscillating metric solutions in
these theories without the addition of other fields.Comment: 15 pages, 1 figur
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