"Science is the poetry of reality"

- Richard Dawkins

In 1974 Stephen Hawking make a prediction that hints a connection between quantum mechanics, general relativity and thermodynamics. A prediction that was apparently doom to never be tested until a theoretical analogue proposed by Bill Unruh in 1980 connected Hawking's prediction with moving media. In recent years, laboratory analogues of the event horizon are getting closer to test one of the most elusive effects in Physics: Hawking radiation. What if we could recreate some aspects of these astrophysical objects in the laboratory? This is the field known as analogue gravity.

The nonlinear propagation of light in media has been a topic of research since the 1980s through the numerical simulation of the non-linear Schrödinger equation (NLSE. On the other hand, thanks to the development of photonic crystal fibers (PCs), ultra-short pulses have become a commonplace in optics labs, these pulses contain only few optical cycles and therefore its theoretical and numerical modeling is still a challenge due to highly nonlinear effects present in them, causing effects such as: soliton pulse formation, optical four-wave mixing and supercontinuum generation.

Quantum mechanics is one of the pillars of modern physics. Then, it comes as a surprise that only a handful of systems can be exactly-solved in quantum theory. All of those can be solved with an algebraic method known as factorization method. Nowadays, a generalization of this method is used to study quantum systems, algebras, differential equations, special functions and coherent states. This method is also known as Darboux transformations, intertwining technique or supersymmetric quantum mechanics.

The special functions play a fundamental role in mathematical physics; some of them are solutions to linear differential equations. In 1905, Paul Painlevé studied the nonlinear ordinary differential equations with specific mathematical properties. He obtained six of these equations, which can be considered as nonlinear analogues of the classical special functions. Lately, there has been an interest in these equations as they appear in several physical applications. Painlevé equations can be solved using algebraic techniques developed in quantum theory.