In optical fiber communications the idea of 'dispersion management' was conceived already in 1980 to create stable pulses in glass fiber cables by periodically varying the dispersion along the cable. It was implemented by Tkach and Chraplyvy in the 1990s. Using the dispersion management idea they created a new type of fiber cable which allowed for ultra high-speed data transfer through optical fibers over intercontinental distances. Record breaking transmission rates of more than 1 Tbits/s over an 18,000 kilometer optical fiber had been achieved using the dispersion management technology. The propagation of pulses through a dispersion managed glass fiber cable is described by the Gabitov-Turitsyn equation, which is a non-local version of the non-linear Schrodinger equation. This equation has been extensively studied numerically and on the level of theoretical physics due to its enormous practical relevance in the modeling of signal-transfer through ultra-high speed glass-fiber cables, but rigorous results are rare. As a test: google 'dispersion management' and you'll get an overwhelming amount of hits (ca 551,000 on google scholar at the moment, much more on plain google) but only very few are rigorous (I know of 6). This is mainly due to the non-local nature of the Gabitov-Turitsyn equation which makes it hard to study. We describe very recent work on the decay and regularity properties of stationary solutions of the Gabitov-Turitsyn equation (the so-called dispersion management solitons). Our results include a simple proof of existence of solutions of the dispersion management equation, regularity of weak solutions and, most recently, a proof of exponential decay of dispersion management solitons. This is joint work with Burak Erdogan and Young-Ran Lee.