Applied and computational complex analysis in the study of nonlinear phenomena

Abstract

The theme issue surveys recent advances at the intersection between Applied and Computational Complex Analysis (ACCA) and the study of different nonlinear phenomena. While complex analysis has traditionally played an important role in the development of certain areas of physics, such as fluid and solid mechanics, in recent years there has been a resurgence of interest in new applications of complex analysis to many different areas. This renaissance has been largely driven by new development in complex analysis which include, but are not limited to, connections with classical, harmonic and asymptotic analysis, new methods based on integral transforms and conformal maps, as well as progress in the Wiener–Hopf and Riemann–Hilbert problems, These advances have been buttressed by parallel development of computational methods reflecting the fact that complex analysis has also come to play an important role in numerical analysis. In fluid mechanics, vortex dynamics continues to drive many new applications of complex analysis. New connections have been discovered between the Riemann–Hilbert problem and integrable hierarchies of partial differential equations (PDEs), Painlevé transcendents as well as problems in statistical mechanics. Rational approximations, orthogonal polynomials, tracking complex singularities in PDEs and conformal mappings have created new opportunities in the study of nonlinear problems.