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On Whitham's conjecture of a highest cusped wave for a nonlocal dispersive equation

Ehrnström, Mats LU and Wahlén, Erik LU (2019) In Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Abstract


We consider the Whitham equation u
t
+2uu
x
+Lu
x
=0, where L is the nonlocal Fourier multiplier operator given by the symbol m(ξ)=tanh⁡ξ/ξ. G. B. Whitham conjectured that for this equation there would be a highest, cusped, travelling-wave solution. We find this wave as a limiting... (More)


We consider the Whitham equation u
t
+2uu
x
+Lu
x
=0, where L is the nonlocal Fourier multiplier operator given by the symbol m(ξ)=tanh⁡ξ/ξ. G. B. Whitham conjectured that for this equation there would be a highest, cusped, travelling-wave solution. We find this wave as a limiting case at the end of the main bifurcation curve of P-periodic solutions, and give several qualitative properties of it, including its optimal C
1/2
-regularity. An essential part of the proof consists in an analysis of the integral kernel corresponding to the symbol m(ξ), and a following study of the highest wave. In particular, we show that the integral kernel corresponding to the symbol m(ξ) is completely monotone, and provide an explicit representation formula for it. Our methods may be generalised.

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author
organization
publishing date
type
Contribution to journal
publication status
epub
subject
keywords
Full-dispersion models, Global bifurcation, Highest waves, Whitham equation
in
Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
publisher
Elsevier
external identifiers
  • scopus:85064243578
ISSN
0294-1449
DOI
10.1016/j.anihpc.2019.02.006
language
English
LU publication?
yes
id
20eacbee-ec3f-4320-aaa9-bacd454ab825
date added to LUP
2019-05-08 12:07:23
date last changed
2019-12-03 02:08:02
@article{20eacbee-ec3f-4320-aaa9-bacd454ab825,
  abstract     = {<p><br>
                                                         We consider the Whitham equation u                             <br>
                            <sub>t</sub><br>
                                                         +2uu                             <br>
                            <sub>x</sub><br>
                                                         +Lu                             <br>
                            <sub>x</sub><br>
                                                         =0, where L is the nonlocal Fourier multiplier operator given by the symbol m(ξ)=tanh⁡ξ/ξ. G. B. Whitham conjectured that for this equation there would be a highest, cusped, travelling-wave solution. We find this wave as a limiting case at the end of the main bifurcation curve of P-periodic solutions, and give several qualitative properties of it, including its optimal C                             <br>
                            <sup>1/2</sup><br>
                                                         -regularity. An essential part of the proof consists in an analysis of the integral kernel corresponding to the symbol m(ξ), and a following study of the highest wave. In particular, we show that the integral kernel corresponding to the symbol m(ξ) is completely monotone, and provide an explicit representation formula for it. Our methods may be generalised.                         <br>
                        </p>},
  author       = {Ehrnström, Mats and Wahlén, Erik},
  issn         = {0294-1449},
  language     = {eng},
  publisher    = {Elsevier},
  series       = {Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire},
  title        = {On Whitham's conjecture of a highest cusped wave for a nonlocal dispersive equation},
  url          = {http://dx.doi.org/10.1016/j.anihpc.2019.02.006},
  doi          = {10.1016/j.anihpc.2019.02.006},
  year         = {2019},
}