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 
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/
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     title = {Levi{\textendash}Civita theory for irrotational water waves in a~one dimensional channel and the complex {Korteweg{\textendash}de~Vries} equation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     url = {http://geodesic.mathdoc.fr/item/TMF_1994_99_3_a11/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] T. Levi-Civita, Rend. Ace. Lincei, 16 (1907), 777–790 | Zbl

[2] U. Cisotti, Rend. Ace. Lincei, 28 (1919), 196–199 | Zbl

[3] U. Crudeli, II Nuovo Cimento, 25 (1923), 43–62 ; Rend. Acc. Lincei, 28 (1919), 174–178; 29 (1920), 265–269; U. Cisotti, Rend. Acc. Lincei, 27 (1918), 255–259, 312–316 ; 29 (1920), 131–133, 175–180, 261–264 | DOI | Zbl | Zbl | Zbl

[4] T. Levi-Civita, Questioni di Meccanica Classica e Relativistica, Zanichelli, Bologna, 1924