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Numerical Simulations of Overturned Traveling Waves

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Nonlinear Water Waves

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Abstract

Dimension-breaking continuation as a numerical technique for computing large amplitude, overturned traveling waves is presented. Dimension-breaking bifurcations from branches of planar waves are presented in two weakly-nonlinear model equations as well as in the vortex sheet formulation of the water wave problem, with the small scale approximation (Ambrose et al., J Comput Phys 247:168–191, 2013; Akers and Reeger, Wave Motion 68:210–217, 2017). The challenges and potential of this method toward computing overturned traveling waves at the interface between three-dimensional fluids is reviewed. Numerical simulations of dimension-breaking continuation are presented in each model. Overturned traveling three-dimensional waves are presented in the vortex sheet system.

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Acknowledgements

Benjamin Akers and Matthew Seiders were supported in part by the Air Force Office of Scientific Research (AFOSR) and the Office of Naval Research (ONR) during the preparation of this manuscript.

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Authors and Affiliations

  1. Air Force Institute of Technology, WPAFB, Dayton, OH, USA

    Benjamin F. Akers & Matthew Seiders

Authors
  1. Benjamin F. Akers
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Correspondence to Benjamin F. Akers .

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Editors and Affiliations

  1. School of Mathematical Sciences, University College Cork, Cork, Ireland

    David Henry

  2. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK

    Konstantinos Kalimeris

  3. School of Mathematics, University of East Anglia, Norwich, UK

    Emilian I. Părău

  4. Department of Mathematics, University College London, London, UK

    Jean-Marc Vanden-Broeck

  5. Centre for Mathematical Sciences, Lund University, Lund, Sweden

    Erik Wahlén

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Akers, B.F., Seiders, M. (2019). Numerical Simulations of Overturned Traveling Waves. In: Henry, D., Kalimeris, K., Părău, E., Vanden-Broeck, JM., Wahlén, E. (eds) Nonlinear Water Waves . Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-33536-6_7

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