25,597 research outputs found
Far-off-resonant wave interaction in one-dimensional photonic crystals with quadratic nonlinearity
We extend a recently developed Hamiltonian formalism for nonlinear wave
interaction processes in spatially periodic dielectric structures to the
far-off-resonant regime, and investigate numerically the three-wave resonance
conditions in a one-dimensional optical medium with nonlinearity.
In particular, we demonstrate that the cascading of nonresonant wave
interaction processes generates an effective nonlinear response in
these systems. We obtain the corresponding coupling coefficients through
appropriate normal form transformations that formally lead to the Zakharov
equation for spatially periodic optical media.Comment: 14 pages, 4 figure
Generalised Elliptic Functions
We consider multiply periodic functions, sometimes called Abelian functions,
defined with respect to the period matrices associated with classes of
algebraic curves. We realise them as generalisations of the Weierstras
P-function using two different approaches. These functions arise naturally as
solutions to some of the important equations of mathematical physics and their
differential equations, addition formulae, and applications have all been
recent topics of study.
The first approach discussed sees the functions defined as logarithmic
derivatives of the sigma-function, a modified Riemann theta-function. We can
make use of known properties of the sigma function to derive power series
expansions and in turn the properties mentioned above. This approach has been
extended to a wide range of non hyperelliptic and higher genus curves and an
overview of recent results is given.
The second approach defines the functions algebraically, after first
modifying the curve into its equivariant form. This approach allows the use of
representation theory to derive a range of results at lower computational cost.
We discuss the development of this theory for hyperelliptic curves and how it
may be extended in the future.Comment: 16 page
Darboux coordinates for the Hamiltonian of first order Einstein-Cartan gravity
Based on preliminary analysis of the Hamiltonian formulation of the first
order Einstein-Cartan action (arXiv:0902.0856 [gr-qc] and arXiv:0907.1553
[gr-qc]) we derive the Darboux coordinates, which are a unique and uniform
change of variables preserving equivalence with the original action in all
spacetime dimensions higher than two. Considerable simplification of the
Hamiltonian formulation using the Darboux coordinates, compared with direct
analysis, is explicitly demonstrated. Even an incomplete Hamiltonian analysis
in combination with known symmetries of the Einstein-Cartan action and the
equivalence of Hamiltonian and Lagrangian formulations allows us to
unambiguously conclude that the \textit{unique} \textit{gauge} invariances
generated by the first class constraints of the Einstein-Cartan action and the
corresponding Hamiltonian are \textit{translation and rotation in the tangent
space}. Diffeomorphism invariance, though a manifest invariance of the action,
is not generated by the first class constraints of the theory.Comment: 44 pages, references are added, organization of material is slightly
modified (additional section is introduced), more details of calculation of
the Dirac bracket between translational and rotational constraints are
provide
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