Numerical Simulations of Overturned Traveling Waves

  • Benjamin F. AkersEmail author
  • Matthew Seiders
Part of the Tutorials, Schools, and Workshops in the Mathematical Sciences book series (TSWMS)


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.


Traveling waves Overturned Numerical continuation 

Mathematics Subject Classification (2000)

35B35 76B15 



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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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Air Force Institute of Technology, WPAFBDaytonUSA

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