  Appearing in 1921 as an equation for small-amplitude waves on the surface of
an infinitely deep liquid, the Nekrasov equation quickly became a source of new
results. This manifested itself both in the field of mathematics (theory of
nonlinear integral equations of A.I. Nekrasov; 1922, later - of N.N. Nazarov;
1941), and in the field of mechanics (transition to a fluid of finite depth -
A.I. Nekrasov; 1927 and refusal on the smallness of the wave amplitude - Yu.P.
Krasovskii; 1960).The main task of the author is to find out the prehistory of
the Nekrasov equation and to trace the change in approaches to its solution in
the context of the nonlinear functional analysis development in the 1940s -
1960s. Close attention will be paid to the contribution of European and Russian
mathematicians and mechanics: A.M. Lyapunov, E. Schmidt, T. Levi-Civita, A.
Villat, L. Lichtenstein, M.A. Krasnoselskii, N.N. Moiseev, V.V. Pokornyi, etc.
In the context of the development of qualitative methods for the Nekrasov
equation investigating, the question of the interaction between Voronezh school
of nonlinear functional analysis under the guidance of Professor M.A.
Krasnoselskii and Rostov school of nonlinear mechanics under the guidance of
Professor I.I. Vorovich.
