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The Benjamin–Lighthill Conjecture for Near-Critical Values of Bernoulli’s Constant

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Abstract

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 Bernoulli’s 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.

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Authors and Affiliations

  1. Department of Mathematics, Linköping University, 581 83, Linköping, Sweden

    Vladimir Kozlov

  2. Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, V.O., Bol’shoy pr. 61, St. Petersburg, 199178, Russian Federation

    Nikolay Kuznetsov

Authors
  1. Vladimir Kozlov
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  2. Nikolay Kuznetsov
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Correspondence to Nikolay Kuznetsov.

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Communicated by C.A. Stuart

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Kozlov, V., Kuznetsov, N. The Benjamin–Lighthill Conjecture for Near-Critical Values of Bernoulli’s Constant. Arch Rational Mech Anal 197, 433–488 (2010). https://doi.org/10.1007/s00205-009-0279-9

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  • DOI: https://doi.org/10.1007/s00205-009-0279-9

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