40 research outputs found
Solitary waves in nonlocal NLS with dispersion averaged saturated nonlinearities
A nonlinear Schr\"odinger equation (NLS) with dispersion averaged
nonlinearity of saturated type is considered. Such a nonlocal NLS is of
integro-differential type and it arises naturally in modeling fiber-optics
communication systems with periodically varying dispersion profile (dispersion
management). The associated constrained variational principle is shown to
posses a ground state solution by constructing a convergent minimizing sequence
through the application of a method similar to the classical concentration
compactness principle of Lions. One of the obstacles in applying this
variational approach is that a saturated nonlocal nonlinearity does not satisfy
uniformly the so-called strict sub-additivity condition. This is overcome by
applying a special version of Ekeland's variational principle.Comment: 24 page
Hausdorff dimension of three-period orbits in Birkhoff billiards
We prove that the Hausdorff dimension of the set of three-period orbits in
classical billiards is at most one. Moreover, if the set of three-period orbits
has Hausdorff dimension one, then it has a tangent line at almost every point.Comment: 10 pages, 1 figur
A Dihedral Acute Triangulation of the Cube
It is shown that there exists a dihedral acute triangulation of the
three-dimensional cube. The method of constructing the acute triangulation is
described, and symmetries of the triangulation are discussed.Comment: Minor edits for journal version. Added some material to the
introductio