Detection of complex singularities for a function of several variables

Kamyar Malakuti, Russel Caflisch, Michael Siegel, Alex Virodov

Research output: Contribution to journalArticle

Abstract

A numerical method for investigating singularities in solutions to non-linear evolution equations is presented. The method is based on a complex analytical approach to singularities introduced by Sulem, Sulem and Frisch, which uses analytic continuation of an independent variable and numerical detection of the width of the analyticity strip, defined as the distance δ from the real domain to the nearest complex singularity. Their method, originally formulated for functions of a single variable, is here generalized to problems and functions of several variables. We first analyse the asymptotic behaviour of the multidimensional Fourier transform of an analytic function, and use this to numerically detect the complex singularity surface. The approach allows us to determine the parameters that characterize the singularity surface in a neighbourhood of its closest point to the real domain.

Original languageEnglish (US)
Pages (from-to)714-728
Number of pages15
JournalIMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)
Volume78
Issue number4
DOIs
StatePublished - Aug 2013

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Several Variables
Singularity
Numerical methods
Fourier transforms
Analytic Continuation
Nonlinear Evolution Equations
Analyticity
Strip
Fourier transform
Analytic function
Asymptotic Behavior
Numerical Methods

Keywords

  • complex singularity
  • form fit
  • Fourier transform

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

Detection of complex singularities for a function of several variables. / Malakuti, Kamyar; Caflisch, Russel; Siegel, Michael; Virodov, Alex.

In: IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications), Vol. 78, No. 4, 08.2013, p. 714-728.

Research output: Contribution to journalArticle

Malakuti, Kamyar ; Caflisch, Russel ; Siegel, Michael ; Virodov, Alex. / Detection of complex singularities for a function of several variables. In: IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications). 2013 ; Vol. 78, No. 4. pp. 714-728.
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