A family of exact solutions approximating the gravitational waves of maximum amplitude. (English. Russian original) Zbl 0888.76009

Karabut, E. A.

Phys.-Dokl. 40, No. 10, 534-537 (1995); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 344, No. 5, 623-626 (1995).

The paper presents a particular family of exact solutions to equations describing finite-amplitude plane periodic waves in an ideal liquid layer of a finite depth over a flat bottom, neglecting the surface tension. The flow is assumed to be irrotational. By means of the appropriately chosen conformal map, the problem is reduced to a nonlinear boundary value problem. In the special case corresponding to the surface waves of the maximum amplitude (with cusp singularities on the free surface), the problem can be reduced to a system of three coupled ODEs that admits a solution in quadratures, in the form of a one-parametric family of periodic waves. The analytical findings are corroborated by comparison with numerical results.

Reviewer: B.A.Malomed (Tel Aviv)

MSC:

76B15

Water waves, gravity waves; dispersion and scattering, nonlinear interaction

Keywords:

conformal map; cusp; ideal liquid layer of finite depth; flat bottom; nonlinear boundary value problem; cusp singularities