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On the problem of a solitary wave on the fluid surface. (English. Russian original) Zbl 0872.76016
Phys.-Dokl. 39, No. 7, 538-541 (1994); translation from Dokl. Akad. Nauk 337, No. 3, 339-341 (1994).
A planar irrotational steady flow of a perfect incompressible heavy fluid over a plane bottom is under consideration. Earlier J. Witting [SIAM J. Appl. Math. 28, 700-719 (1975; Zbl 0297.76015)] constructed a solution \(f\) for the solitary wave as a formal power series. Now the author finds the exact sum of the Witting series. It is shown that such a series does not describe the solitary wave, but corresponds to certain other steady flows of heavy fluid with free boundary. The found solutions are close to solitary waves, degree of closeness increasing with a decrease of the amplitude. The comparison with numerical results of Witting shows that he considered an insufficient number of series’ terms.

76B25 Solitary waves for incompressible inviscid fluids