Ultrashort pulses are severely distorted even by low dispersive media. While the mathematical analysis of dispersion is well known, the technical literature focuses on pulses, which keep their shape: Gaussian and Airy pulses. However, the cases where the shape of the pulse is unaffected by dispersion is the exception rather than the norm. It is the object of this chapter to present a variety of pulses profiles, which have analytical expressions but can simulate real-physical pulses with greater accuracy. In particular, the dynamics of smooth rectangular pulses, physical Nyquist-Sinc pulses, and slowly rising but sharply decaying ones (and vice-versa) are presented. Besides the usage of this chapter as a handbook of analytical expressions for pulses' propagations in a dispersive medium, there are several new findings. The main ones are: Analytical expressions for the propagation of chirped rectangular pulses, which converge to extremely short pulses; analytical approximation for the propagation of Super-Gaussian pulses; the propagation of Nyquist Sinc Pulse with smooth spectral boundaries and an analytical expression for a physical realization of an attenuation compensating Airy pulse.

Schrödinger dynamics is a nonlocal process. Not only does local perturbation affect instantaneously the entire space, but the effect decays slowly. When the wavefunction is spectrally bounded, the Schrödinger equation can be written as a universal set of ordinary differential equations, with universal coupling between them, which is related to Euler’s formula. Since every variable represents a different local value of the wave equation, the coupling represents the dynamics’ nonlocality. It is shown that the nonlocal coefficient is inversely proportional to the distance between the centers of these local areas. As far as we know, this is the first time that this inverse square law was formulated.

The scattering pattern from a narrow absorbing/amplifying grating is investigated. A simple model of a narrow amplifying grating is solved exactly numerically and approximately analytically for the regime where the beam’s wavelength is much shorter than the grating’s wavelength. The main results are: The incident angle divides the scattering pattern into two regimes: below and above the incident angles. The former regime has a weak dependence on the incident angle but has a strong dependence on the scattering one. In this regime, a new grating formula is derived. The opposite occurs in the latter regime, which is very sensitive to the incident angle, but has only weak dependence on the scattering angle. Consequently, at certain incident angles, all the scattering is concentrated in the first regime, i.e., all scattering angles are lower than the incident angle.