Higher-order approximations of cnoidal-wave theory

A solution of the problem of gravity waves on a liquid surface is sought in the form of a series whose first term corresponds to shallow-water theory. Such series have been previously studied numerically and analytically but their structure remains unclear because of the complicated initial formulation of the problem. In the present paper, instead of the strongly linear boundary-valua problem with a free boundary containing several unknown functions, we solve an ordinary quadratic-nonlinear differential-difference equation of the first order containing an unknown function.