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Gibbs phenomenon for dispersive PDEs on the line. (English) Zbl 1368.35065

Summary: 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.

MSC:

35C20 Asymptotic expansions of solutions to PDEs
35C06 Self-similar solutions to PDEs
41A55 Approximate quadratures
35E15 Initial value problems for PDEs and systems of PDEs with constant coefficients

Software:

RHPackage; DLMF
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Full Text: DOI arXiv

References:

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