Russel E. Caflisch^[1], Francesco Gargano^[2], Marco Sammartino^[2] and Vincenzo Sciacca^[2]

Complex singularities and PDEs

Pages: 69-133
Received: 13 February 2015   
Accepted in revised form: 8 May 2015.
Mathematics Subject Classification (2010): 35(35A20 35Q 35Q35 35Q53), 65(65M 65M60), 76(76D05 76D10).
Keywords: Complex singularity, Fourier transforms, Padé approximation, Borel and power series methods, dispersive shocks, fluid mechanics, zero viscosity.
Author address:
[1] : Mathematics Department, University of California at Los Angeles, Los Angeles, CA 90036, United States
[2] : Dipartimento di Matematica, Università di Palermo, 90123 Palermo, Italy

The work of FG, MS and VS has been partially supported by the GNFM of INDAM.

Abstract: 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.

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