We study traveling waves for the vortex sheet with surface tension. We use the angle-arclength description of the interface rather than Cartesian coordinates, and we utilize an arclength parameterization as well. In this setting, we make a new formulation of the traveling wave ansatz. For this problem, it should be possible for traveling waves to overturn, and notably, our formulation does allow for waves with multi-valued height. We prove that there exist traveling vortex sheets with surface tension bifurcating from equilibrium. We compute these waves by means of a quasi-Newton iteration in Fourier space; we find continua of traveling waves bifurcating from equilibrium and extending to include overturning waves, for a variety of values of the mean vortex sheet strength.
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Benjamin Akers, David M. Ambrose, J. Douglas Wright, Traveling waves from the arclength parameterization: Vortex sheets with surface tension. Interfaces Free Bound. 15 (2013), no. 3, pp. 359–380DOI 10.4171/IFB/306