Gibbs phenomenon for dispersive PDEs on the line

Gino Biondini; Thomas Trogdon

doi:10.1137/16M1090892

Gibbs phenomenon for dispersive PDEs on the line

Gino Biondini
, Thomas Trogdon

Mathematics

University at Buffalo

University of California at Irvine

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

Abstract

We investigate the Cauchy problem for linear, constant-coefficient evolution PDEs on the real line with discontinuous initial conditions (ICs) in the small-time limit. The small-time behavior of the solution near discontinuities is expressed in terms of universal, computable special functions. We show that the leading-order behavior of the solution of dispersive PDEs near a discontinuity of the ICs is characterized by Gibbs-type oscillations and gives exactly the Wilbraham-Gibbs constants.

Original language	English
Pages (from-to)	813-837
Number of pages	25
Journal	SIAM Journal on Applied Mathematics
Volume	77
Issue number	3
DOIs	https://doi.org/10.1137/16M1090892
State	Published - 2017

Keywords

Asymptotic expansions
Dispersive PDEs
Gibbs phenomenon
Steepest descent

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10.1137/16M1090892

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abstract = "We investigate the Cauchy problem for linear, constant-coefficient evolution PDEs on the real line with discontinuous initial conditions (ICs) in the small-time limit. The small-time behavior of the solution near discontinuities is expressed in terms of universal, computable special functions. We show that the leading-order behavior of the solution of dispersive PDEs near a discontinuity of the ICs is characterized by Gibbs-type oscillations and gives exactly the Wilbraham-Gibbs constants.",

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AU - Trogdon, Thomas

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AB - We investigate the Cauchy problem for linear, constant-coefficient evolution PDEs on the real line with discontinuous initial conditions (ICs) in the small-time limit. The small-time behavior of the solution near discontinuities is expressed in terms of universal, computable special functions. We show that the leading-order behavior of the solution of dispersive PDEs near a discontinuity of the ICs is characterized by Gibbs-type oscillations and gives exactly the Wilbraham-Gibbs constants.

KW - Asymptotic expansions

KW - Dispersive PDEs

KW - Gibbs phenomenon

KW - Steepest descent

UR - https://www.scopus.com/pages/publications/85022069826

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DO - 10.1137/16M1090892

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JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

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ER -

Gibbs phenomenon for dispersive PDEs on the line

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Keywords

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