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Poles and Branch Cuts in Free Surface Hydrodynamics

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Abstract

We consider the motion of ideal incompressible fluid with free surface. We analyzed the exact fluid dynamics through the time-dependent conformal mapping \(z=x+iy=z(w,t)\) of the lower complex half plane of the conformal variable w into the area occupied by fluid. We established the exact results on the existence vs. nonexistence of the pole and power law branch point solutions for \(1/z_w\) and the complex velocity. We also proved the nonexistence of the time-dependent rational solution of that problem for the second- and the first-order moving pole.

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Acknowledgements

We thank the support of the Russian Ministry of Science and Higher Education. The work of P.M.L. was supported by the National Science Foundation, Grant DMS-1814619. The work of V.E.Z. was supported by the National Science Foundation, Grant number DMS-1715323. The work of V.E.Z. described in Sect. 2 was supported by the Russian Science Foundation, Grant number 19-72-30028. Simulations were performed at the Texas Advanced Computing Center using the Extreme Science and Engineering Discovery Environment (XSEDE), supported by NSF Grant ACI-1053575. P.M.L. would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the program “Complex analysis: techniques, applications and computations” where work on this paper was partially undertaken.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM, 87131, USA

    P. M. Lushnikov

  2. Landau Institute for Theoretical Physics, Chernogolovka, 142432, Russia

    P. M. Lushnikov & V. E. Zakharov

  3. Center for Advanced Studies, Skoltech, Moscow, 143026, Russia

    V. E. Zakharov

  4. Department of Mathematics, University of Arizona, Tucson, AZ, 85721, USA

    V. E. Zakharov

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  1. P. M. Lushnikov
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Lushnikov, P.M., Zakharov, V.E. Poles and Branch Cuts in Free Surface Hydrodynamics. Water Waves 3, 251–266 (2021). https://doi.org/10.1007/s42286-020-00040-y

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