A tale of two Nekrasov's integral equations

TL;DR

This paper explores the historical development, derivation, and solution approaches of Nekrasov's integral equations for steady water waves, highlighting their significance and Nekrasov's personal history.

Contribution

It provides a detailed analysis of Nekrasov's original derivation, solution methods, and subsequent developments related to his integral equations.

Findings

Historical insights into Nekrasov's work and life

Survey of solution techniques for Nekrasov's equations

Discussion of related mathematical topics

Abstract

Just 100 years ago, Nekrasov published the widely cited paper \cite{N1}, 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 this connection, Nekrasov's life after 1917 is briefly outlined, in particular, how he became a victim of Stalin's terror. Further results concerning Nekrasov's equations and related topicz are surveyed.