23 research outputs found
Meromorphic traveling wave solutions of the complex cubic-quintic Ginzburg-Landau equation
We look for singlevalued solutions of the squared modulus M of the traveling
wave reduction of the complex cubic-quintic Ginzburg-Landau equation. Using
Clunie's lemma, we first prove that any meromorphic solution M is necessarily
elliptic or degenerate elliptic. We then give the two canonical decompositions
of the new elliptic solution recently obtained by the subequation method.Comment: 14 pages, no figure, to appear, Acta Applicandae Mathematica
Are Devaney hairs fast escaping?
Beginning with Devaney, several authors have studied transcendental entire
functions for which every point in the escaping set can be connected to
infinity by a curve in the escaping set. Such curves are often called Devaney
hairs. We show that, in many cases, every point in such a curve, apart from
possibly a finite endpoint of the curve, belongs to the fast escaping set. We
also give an example of a Devaney hair which lies in a logarithmic tract of a
transcendental entire function and contains no fast escaping points.Comment: 22 pages, 1 figur
Entire functions with Julia sets of positive measure
Let f be a transcendental entire function for which the set of critical and
asymptotic values is bounded. The Denjoy-Carleman-Ahlfors theorem implies that
if the set of all z for which |f(z)|>R has N components for some R>0, then the
order of f is at least N/2. More precisely, we have log log M(r,f) > (N/2) log
r - O(1), where M(r,f) denotes the maximum modulus of f. We show that if f does
not grow much faster than this, then the escaping set and the Julia set of f
have positive Lebesgue measure. However, as soon as the order of f exceeds N/2,
this need not be true. The proof requires a sharpened form of an estimate of
Tsuji related to the Denjoy-Carleman-Ahlfors theorem.Comment: 17 page
Whispering-gallery modes in shielded hemispherical dielectric resonators
The results of the numerical and experimental investigations of whispering-gallery (WG) modes in shielded hemispherical dielectric resonators are presented in this paper. It is shown that the Q factor of WG modes in the shielded resonator can be ten times much higher than the Q factor of the similar open hemispherical dielectric-resonator modes. Shielding the resonator can decrease the dimensions of both the dielectric hemisphere and resonator as a whole, saving the high-Q factor of WG modes. The usage of a cylindrical shield and local flat reflectors in the experiment provides the investigation of the high-Q factor of WG modes in the resonator