Relativistic analogue in non-paraxial shock waves

Shock generation and wave-breaking are effects largely investigated in nonlinear optics. They are always occurring in extreme regimes with a variety of fundamental physical implications, and a number of applications, ranging from particle and material manipulation, to supercontinuum and X-ray generation.

In nonlinear optics, one studies shocks in space and time. Concerning the spatial case, shock waves are observed as highly irregular wave-fronts that originate upon the propagation of a smooth Gaussian beam in a strongly nonlinear medium like, for example, a thermal liquid.

So far, the analysis of spatial shock waves has been limited by the paraxial approximation. The validity of this approximation, however, is questioned by the large spatial bandwidth that is observed at the shock formation.

In a manuscript published in Optics Communications (arXiv:1412.8602) Silvia Gentilini, Eugenio Del Re and Claudio Conti study theoretically and computationally the effects of the non-paraxial regime on the shocks. The result is a predicted correction to the maximal spatial bandwidth after the shock generation, which is within experimentally measurable values, and which is also relevant for temporal shock waves.

The analysis is fascinating, as it shows that non-paraxial terms can be mapped to the relativistic corrections that occur when one considers the propagation of particles with velocity comparable with the speed of light. In other words, the mathematical treatment of the non-paraxial shock is analogue to the treatment of the relativistic particle motion. The trajectories of the particles corresponds to the so-called characteristic lines.

This problem is also relevant for mathematical investigations concerning wave-breaking in high-order nonlinear partial differential equations.

The picture below shows an example of  the calculation of the relativistic shock wave front.