141 research outputs found
Isotropic realizability of current fields in R^3
This paper deals with the isotropic realizability of a given regular
divergence free field j in R^3 as a current field, namely to know when j can be
written as sigma Du for some isotropic conductivity sigma, and some gradient
field Du. The local isotropic realizability in R^3 is obtained by Frobenius'
theorem provided that j and curl j are orthogonal in R^3. A counter-example
shows that Frobenius' condition is not sufficient to derive the global
isotropic realizability in R^3. However, assuming that (j, curl j, j x curl j)
is an orthogonal basis of R^3, an admissible conductivity sigma is constructed
from a combination of the three dynamical flows along the directions j/|j|,
curl j/|curl j| and (j/|j|^2) x curl j. When the field j is periodic, the
isotropic realizability in the torus needs in addition a boundedness assumption
satisfied by the flow along the third direction (j/|j|^2) x \curl j. Several
examples illustrate the sharpness of the realizability conditions.Comment: 22 page
Field patterns: a new type of wave with infinitely degenerate band structure
Field pattern materials (FP-materials) are space-time composites with
PT-symmetry in which the one-dimensional- spatial distribution of the
constituents changes in time in such a special manner to give rise to a new
type of waves, which we call field pattern waves (FP-waves) [G. W. Milton and
O. Mattei, Proc. R. Soc. A 473, 20160819 (2017), O. Mattei and G. W. Milton,
arXiv:1705.00539 (2017)]. Specifically, due to the special periodic space-time
geometry of these materials, when an instantaneous disturbance propagates
through the system, the branching of the characteristic lines at the space-time
interfaces between phases does not lead to a chaotic cascade of disturbances
but concentrates on an orderly pattern of disturbances: this is the field
pattern. By applying Bloch-Floquet theory we find that the dispersion diagrams
associated with these FP-materials are infinitely degenerate: associated with
each point on the dispersion diagram is an infinite space of Bloch functions, a
basis for which are generalized functions each concentrated on a field pattern,
paramaterized by a variable that we call the launch parameter. The dynamics
separates into independent dynamics on the different field patterns, each with
the same dispersion relation.Comment: 8 pages, 4 figure
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