Geodesic
Levi--Civita theory for irrotational water waves in a~one dimensional channel and the complex Korteweg--de~Vries equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 99 (1994) no. 3, pp. 435-440

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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. 
@article{TMF_1994_99_3_a11,
     author = {D. Levi},
     title = {Levi--Civita theory for irrotational water waves in a~one dimensional channel and the complex {Korteweg--de~Vries} equation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {435--440},
     publisher = {mathdoc},
     volume = {99},
     number = {3},
     year = {1994},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1994_99_3_a11/}
}
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D. Levi. Levi--Civita theory for irrotational water waves in a~one dimensional channel and the complex Korteweg--de~Vries equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 99 (1994) no. 3, pp. 435-440. http://geodesic.mathdoc.fr/item/TMF_1994_99_3_a11/