Levi–Civita theory for irrotational water waves in a one dimensional channel and the complex Korteweg–de Vries equation

We review Levi–Civita theory that 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 that, for almost horizontal flow and for initial conditions independent from the vertical variable, reduces to the usual one.