Athanassios S. Fokas

Bio: Athanassios S. Fokas is an academic researcher from University of Cambridge. The author has contributed to research in topics: Boundary value problem & Nonlinear system. The author has an hindex of 59, co-authored 275 publications receiving 14467 citations. Previous affiliations of Athanassios S. Fokas include Clarkson College & University of Paderborn.

The isomonodromy approach to matric models in 2D quantum gravity

Abstract: We consider the double-scaling limit in the hermitian matrix model for 2D quantum gravity associated with the measure exp\(\sum\limits_{j = 1}^N {t_{j^{Z^{2j,} } } N \geqq 3} \) We show that after the appropriate modification of the contour of integration the Cross-Migdal-Douglas-Shenker limit to the Painleve I equation (in the generic case of the pure gravity) is valid and calculate the nonperturbative parameters of the corresponding Painleve function Our approach is based on the WKB-analysis of the L-A pair corresponding to the discrete string equation in the framework of the Inverse Monodromy Method Here we extend our results, which were obtained before for the particular casesN=2,3 Our analysis complements the isomonodromy approach proposed by G Moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself

Complex Variables: Introduction and Applications

Abstract: Complex variables provide powerful methods for attacking problems that can be very difficult to solve in any other way, and it is the aim of this book to provide a thorough grounding in these methods and their application. Part I of this text provides an introduction to the subject, including analytic functions, integration, series, and residue calculus and also includes transform methods, ODEs in the complex plane, and numerical methods. Part II contains conformal mappings, asymptotic expansions, and the study of Riemann–Hilbert problems. The authors provide an extensive array of applications, illustrative examples and homework exercises. This 2003 edition was improved throughout and is ideal for use in undergraduate and introductory graduate level courses in complex variables.

A unified transform method for solving linear and certain nonlinear PDEs

Abstract: A new transform method for solving initial boundary value problems for linear and for integrable nonlinear PDEs in two independent variables is introduced. This unified method is based on the fact that linear and integrable nonlinear equations have the distinguished property that they possess a Lax pair formulation. The implementation of this method involves performing a simultaneous spectral analysis of both parts of the Lax pair and solving a Riemann–Hilbert problem. In addition to a unification in the method of solution, there also exists a unification in the representation of the solution. The sine–Gordon equation in light–cone coordinates, the nonlinear Schrodinger equation and their linearized versions are used as illustrative examples. It is also shown that appropriate deformations of the Lax pairs of linear equations can be used to construct Lax pairs for integrable nonlinear equations. As an example, a new Lax pair of the nonlinear Schrodinger equation is derived.

Controlling Electromagnetic Fields

Abstract: Using the freedom of design that metamaterials provide, we show how electromagnetic fields can be redirected at will and propose a design strategy. The conserved fields-electric displacement field D, magnetic induction field B, and Poynting vector B-are all displaced in a consistent manner. A simple illustration is given of the cloaking of a proscribed volume of space to exclude completely all electromagnetic fields. Our work has relevance to exotic lens design and to the cloaking of objects from electromagnetic fields.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Abstract: to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

The Painlevé property for partial differential equations

Abstract: In this paper we define the Painleve property for partial differential equations and show how it determines, in a remarkably simple manner, the integrability, the Backlund transforms, the linearizing transforms, and the Lax pairs of three well‐known partial differential equations (Burgers’ equation, KdV equation, and the modified KdV equation). This indicates that the Painleve property may provide a unified description of integrable behavior in dynamical systems (ordinary and partial differential equations), while, at the same time, providing an efficient method for determining the integrability of particular systems.