Archive for Rational Mechanics and Analysis

, Volume 228, Issue 3, pp 773–820 | Cite as

A Variational Reduction and the Existence of a Fully Localised Solitary Wave for the Three-Dimensional Water-Wave Problem with Weak Surface Tension

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Abstract

Fully localised solitary waves are travelling-wave solutions of the three- dimensional gravity–capillary water wave problem which decay to zero in every horizontal spatial direction. Their existence has been predicted on the basis of numerical simulations and model equations (in which context they are usually referred to as ‘lumps’), and a mathematically rigorous existence theory for strong surface tension (Bond number \({\beta}\) greater than \({\frac{1}{3}}\)) has recently been given. In this article we present an existence theory for the physically more realistic case \({0 < \beta < \frac{1}{3}}\). A classical variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle featuring a perturbation of the functional associated with the Davey–Stewartson equation. A nontrivial critical point of the reduced functional is found by minimising it over its natural constraint set.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Section de mathématiques, Station 8Ecole Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.Fachrichtung MathematikUniversität des SaarlandesSaarbrückenGermany
  3. 3.Department of Mathematical SciencesLoughborough UniversityLoughboroughUK
  4. 4.Centre for Mathematical SciencesLund UniversityLundSweden

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