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The Benjamin-Lighthill Conjecture for Steady Water Waves (revisited)
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Russian Academy of Science.
2011 (English)In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 201, no 2, 631-645 p.Article in journal (Refereed) Published
Abstract [en]

The two-dimensional nonlinear problem of steady gravity waves on water of finite depth is considered. The Benjamin-Lighthill conjecture is proved for these waves provided Bernoullis constant attains near-critical values. In fact this is a consequence of the following more general results. If Bernoullis constant is near-critical, then all corresponding waves have sufficiently small heights and slopes. Moreover, for every near-critical value of Bernoullis constant, there exist only the following waves: a solitary wave and the family of Stokes waves having their crests strictly below the crest of this solitary wave; this family is parametrised by wave heights which increase from zero to the height of the solitary wave. All these waves are unique up to horizontal translations. Most of these results were proved in our previous paper (Kozlov and Kuznetsov in Arch Rational Mech Anal 197, 433-488, 2010), in which it was supposed that wave slopes are bounded a priori. Here we show that the latter condition is superfluous by proving the following theorem. If any steady wave has the free-surface profile of a sufficiently small height, then the slope of this wave is also small.

Place, publisher, year, edition, pages
Springer Science Business Media , 2011. Vol. 201, no 2, 631-645 p.
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URN: urn:nbn:se:liu:diva-69772DOI: 10.1007/s00205-011-0397-zISI: 000292970700007OAI: diva2:433613
Available from: 2011-08-10 Created: 2011-08-08 Last updated: 2011-08-10

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