Complex singularities and PDEs

In this paper we give a review on the computational methods used to capture and characterize the complex singularities developed by some relevant PDEs. We begin by reviewing the classical singularity tracking method and give an example of application using the Burgers equation as a case study. This method is based on the analysis of the Fourier spectrum of the solution and it allows to determine and characterize the complex singularity closest to the real domain. We then introduce other methods generally used to detect the hidden singularities. In particular we show some applications of the Padé approximation, of the Kida method, and of Borel-Polya method. We apply these techniques to the study of the singularity formation of some nonlinear dispersive and dissipative one dimensional PDE, of the 2D Prandtl equation and of the 2D Kadomtsev-Petviashvili equation. Finally the complex singularity analysis is applied to viscous high Reynolds number incompressible flows in the case of interaction with a rigid wall, and in the case of the vortex layers.