In this paper we develop an existence theory for the Cauchy problem to stochastic Hunter-Saxton equatio, and prove several properties of blow-up its solutions. An important part is continuation solutions equations beyond (wave-breaking). linear noise case, using method (stochastic) characteristics, also study random wave-breaking effects unobserved in deterministic problem. Notably, derive expl...

The state of the art for solving the phase retrieval problem in two dimensions relies heavily on the algorithms proposed by Gerchberg, Saxton, and Fienup. Despite the widespread use of these algorithms, current mathematical theory cannot explain their remarkable success. It is already known that the Gerchberg-Saxton algorithm is a nonconvex version of method of alternating projections. In this ...

We use a three-dimensional Gerchberg–Saxton algorithm (Shabtay (2003) Opt. Commun. 226 33) to calculate the Fourier-space representation of physically realizable light beams with arbitrarily shaped three-dimensional intensity distributions. From this representation we extract a phase-hologram pattern that allows us to create such light beams experimentally. We show several examples of experimen...

In this article we study Sobolev metrics of order one on diffeomorphism groups on the real line. We prove that the space Diff1(R) equipped with the homogenous Sobolev metric of order one is a flat space in the sense of Riemannian geometry, as it is isometric to an open subset of a mapping space equipped with the flat L2-metric. Here Diff1(R) denotes the extension of the group of all either comp...

Two spatial light modulators (SLMs) are utilized for beam splitting, steering and tracking. Both linear and holographic phase screens are used in a demonstration of technology to allow real time tracking to communicate in a one-to-several type scenario. One SLM is used to apply a linear phase modulation to steer multiple beams onto a detector. The spots that are produced represent the targets a...

We show that two families of equations, the generalized inviscid Proudman–Johnson equation and r-Hunter–Saxton (recently introduced by Cotter et al.), coincide for a certain range parameters. This gives new geometric interpretation these equations as geodesic right invariant homogeneous $$W^{1,r}$$ -Finsler metrics on diffeomorphism group. Generalizing construction Lenells Hunter–Saxton equatio...