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A Tale of Two Nekrasov’s Integral Equations

Abstract

One hundred years ago, Nekrasov published the widely cited paper (Nekrasov in Izvestia Ivanovo-Voznesensk Politekhn Inst 3:52–65, 1921), in which he derived the first of his two integral equations describing steady periodic waves on the free surface of water. We examine how Nekrasov arrived at these equations and his approach to investigating their solutions. In connection with this, Nekrasov’s life after 1917 is briefly outlined, in particular, how he became a prisoner in Stalin’s Gulag. Further results concerning Nekrasov’s equations and related topics are surveyed.

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Data availability statement

This article contains two quotations and three figures, all of which are in the public domain. The quotation from A Tale of Two Cities by Charles Dickens is available at https://en.wikipedia.org/wiki/A_Tale_of_Two_Cities. The second quotation is from Stokes’s Appendix to [67] published in the book [68]; it is available at https://archive.org/details/mathphyspapers01stokrich/page/n9/mode/2up. Figure 1 is available at http://ispu.ru/files/imagecache/640x480/cck-images/Prepodavateli_1922.jpg, while Figures 2 and 3 are scans from the books [53] and [52], which were published by the Soviet Academy of Sciences during the period when there was no copyright law in the USSR.

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Acknowledgements

The author is indebted to J. F. Toland for his comments on the presentation in Section 4.

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On the occasion of the centenary of Nekrasov’s equation for deep water.

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Kuznetsov, N. A Tale of Two Nekrasov’s Integral Equations. Water Waves (2021). https://doi.org/10.1007/s42286-021-00051-3

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Keywords

  • Nekrasov’s integral equation
  • Stokes wave
  • Solitary wave
  • Bifurcation theory