Nonlinear system of Euler--Lagrange equations. Reduction to the Korteweg--de Vries equation and periodic solutions

The Euler-Lagrange equations, which correspond to a variational principle with a Lagrange function depending on arbitrary functions and their first order derivatives, are shown to be reducible to the Korteweg-de Vries equation under a small--but finite--amplitude approximation. Closed form periodic solutions to the Euler-Lagrange equations are found for a particular case, and the modulational stability of these solutions is discussed. Equations for waves in cold plasma are discussed as examples