On Whitham's conjecture of a highest cusped wave for a nonlocal dispersive equation
(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)=tanhξ/ξ. G. B. Whitham conjectured that for this equation there would be a highest, cusped, travellingwave solution. We find this wave as a limiting... (More)
(Less)
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)=tanhξ/ξ. G. B. Whitham conjectured that for this equation there would be a highest, cusped, travellingwave solution. We find this wave as a limiting case at the end of the main bifurcation curve of Pperiodic 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.
 author
 Ehrnström, Mats ^{LU} and Wahlén, Erik ^{LU}
 organization
 publishing date
 2019
 type
 Contribution to journal
 publication status
 epub
 subject
 keywords
 Fulldispersion 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
 02941449
 DOI
 10.1016/j.anihpc.2019.02.006
 language
 English
 LU publication?
 yes
 id
 20eacbeeec3f4320aaa9bacd454ab825
 date added to LUP
 20190508 12:07:23
 date last changed
 20191203 02:08:02
@article{20eacbeeec3f4320aaa9bacd454ab825, 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)=tanhξ/ξ. G. B. Whitham conjectured that for this equation there would be a highest, cusped, travellingwave solution. We find this wave as a limiting case at the end of the main bifurcation curve of Pperiodic 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 = {02941449}, 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}, }