An area of intense research is that of photonics, where light propagation features are controlled by clever engineering of periodic optical structures. Perhaps the best known photonic devices are the fiber Bragg grating and photonic crystal fibers. In both cases there is plenty of experimental work that illustrates the rich dynamics that emerges when linear periodic properties coexist with nonlinear intensity dependent effects. Phenomena such as slow gap soliton dynamics in fibers and supercontinuum generation in photonic crystal fibers have been observed only when periodicity and nonlinearity are present.

In this presentation we briefly showcase the multitude of phenomena in photonics, we then concentrate on three nonlinear periodic geometries: a two dimensional nonlinear waveguide Bragg grating a photonic crystal coupler and a periodic array of nonlinear optical fibers. What we will show in the first case is the existence of nontrivial optical bullet dynamics such as light trapping, bending and switching. The second geometry presents a novel system to produced controlled optical pulse delays. Finally the fiber array presents a nonlinear system to study light localization in a discrete disordered system.

Eigenvalues of the Laplacian on triangular domains cannot be computed exactly, in general. But the triangles that extremize the first eigenvalue (the fundamental tone of the membrane) often turn out to be equilateral, or degenerate in some way. These special triangles give sharp eigenvalue bounds for the general case.

Among all triangles with fixed diameter, we prove the degenerate acute isosceles triangle minimizes the Neumann fundamental tone. In the other direction, if we fix perimeter (or area) then the equilateral triangle maximizes the Neumann fundamental tone. Our approach involves variational principles and geometric transformations of the domain, and relies on the explicit formulas for eigenfunctions of equilateral triangles and circular sectors. We also prove symmetry/antisymmetry for eigenfunctions of isosceles triangles.

The set of continuous viscosity solutions of the infinity Laplace
equation $-\bigtriangleup^N_{\infty}w(x) = f(x)$ with generally
sign-changing right-hand-side $f$ in a bounded domain is analyzed.
The existence of a least and a greatest continuous viscosity
solutions up to the boundary is proved through a Perron's
construction by means of a strict comparison principle. These
extremal solutions are proved to be absolutely extremal solutions.