We review the Levi-Civita theory, which reduces the study of the irrotational flow in a one-dimensional channel or the solution of a non-linear differential-functional partial differential equation for the velocity potential. We show how, by considering small perturbations in a shallow water channel, we can reduce the non-linear differential-functional equation to a complex Korteweg-de Vries equation which, for almost horizontal flow and for initial conditions independent of the vertical variable, reduces to the usual one.
Levi, D. (1994). LEVI-CIVITA THEORY FOR IRROTATIONAL WATER-WAVES IN A ONE-DIMENSIONAL CHANNEL AND THE COMPLEX KORTEWEG-DE VRIES EQUATION. THEORETICAL AND MATHEMATICAL PHYSICS, 99(3), 705-709 [10.1007/BF01017056].