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The Benjamin-Lighthill Conjecture for Near-Critical Values of Bernoullis Constant
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Russian Academy of Science.
2010 (English)In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 197, no 2, 433-488 p.Article in journal (Refereed) Published
Abstract [en]

In 1954, Benjamin and Lighthill made a conjecture concerning the classical nonlinear problem of steady gravity waves on water of finite depth. According to this conjecture, a point of some cusped region on the (r, s)-plane (r and s are the non-dimensional Bernoullis constant and the flow force, respectively), corresponds to every steady wave motion described by the problem. Conversely, at least one steady flow corresponds to every point of the region. In the present paper, this conjecture is proved for near-critical flows (when r attains values close to one), under the assumption that the slopes of wave profiles are bounded. Another question studied here concerns the uniqueness of solutions, and it is proved that for every near-critical value of r only the following waves do exist: (i) a unique (up to translations) solitary wave; (ii) a family of Stokes waves (unique up to translations), which is parametrised by the distance from the bottom to the wave crest. The latter parameter belongs to the interval bounded below by the depth of the subcritical uniform stream and above by the distance from the bottom to the crest of solitary wave corresponding to the chosen value of r.

Place, publisher, year, edition, pages
Springer Science Business Media , 2010. Vol. 197, no 2, 433-488 p.
National Category
URN: urn:nbn:se:liu:diva-58335DOI: 10.1007/s00205-009-0279-9ISI: 000279077600003OAI: diva2:343375
Available from: 2010-08-13 Created: 2010-08-11 Last updated: 2010-08-13

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