### 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 language | English (US) |
---|---|

Pages (from-to) | 714-728 |

Number of pages | 15 |

Journal | IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications) |

Volume | 78 |

Issue number | 4 |

DOIs | |

State | Published - Aug 2013 |

### Fingerprint

### Keywords

- complex singularity
- form fit
- Fourier transform

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)*,

*78*(4), 714-728. https://doi.org/10.1093/imamat/hxt017

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

Research output: Contribution to journal › Article

*IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)*, vol. 78, no. 4, pp. 714-728. https://doi.org/10.1093/imamat/hxt017

}

TY - JOUR

T1 - Detection of complex singularities for a function of several variables

AU - Malakuti, Kamyar

AU - Caflisch, Russel

AU - Siegel, Michael

AU - Virodov, Alex

PY - 2013/8

Y1 - 2013/8

N2 - 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.

AB - 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.

KW - complex singularity

KW - form fit

KW - Fourier transform

UR - http://www.scopus.com/inward/record.url?scp=84880532254&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84880532254&partnerID=8YFLogxK

U2 - 10.1093/imamat/hxt017

DO - 10.1093/imamat/hxt017

M3 - Article

AN - SCOPUS:84880532254

VL - 78

SP - 714

EP - 728

JO - IMA Journal of Applied Mathematics

JF - IMA Journal of Applied Mathematics

SN - 0272-4960

IS - 4

ER -