## An improved theory of long waves on water surface.(English. Russian original)Zbl 0881.76016

J. Appl. Math. 61, No. 2, 177-182 (1997); translation from Prikl. Mat. Mekh. 61, No. 2, 184-189 (1997).
The authors consider potential motion of water over infinite horizontal bottom. The water is considered to be an ideal incompressible homogeneous fluid with the depth $$h$$ in unperturbed state. By means of Hamiltonian formalism the equations are derived which describe long waves and take into account the second order terms with respect to the small nonlinearity and dispersion parameters. A relationship between the obtained equations and Korteweg-de Vries equations is established, and $$N$$-soliton solutions are described among exact solutions to the Korteweg-de Vries equations.

### MSC:

 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76B25 Solitary waves for incompressible inviscid fluids

### References:

 [1] Laitone, E. V.: The second approximation to conoidal and solitary waves. J. fluid mech. 9, No. 3, 430-444 (1960) · Zbl 0095.22302 [2] Yegorov, Yu.A; Molotkov, I. A.: On the influence of variable depth and also of non-linear and dispersion terms of the second order on the propagation of gravitational waves. Problemy mat. Fiz. 2, 113-124 (1986) · Zbl 0599.76027 [3] Malykh, A. A.; Seregin, I. A.: Higher approximations in the theory of long waves on the surface of a heavy liquid. Modelirovaniye v mekhani 2(19), No. 6, 77-82 (1988) · Zbl 0703.76017 [4] Arsen’yev, S. A.: On the theory of long waves on water. Dokl. ross. Akad. nauk 334, No. 5, 635-638 (1994) [5] Arsen’yev, S. A.; Vakhrushev, M. M.; Shelkovnikov, N. K.: A new evolution equation for non-linear long waves on water. Vestnik mosk GoS univ. Ser. 3: fizika, astronomiya 36, No. 2, 74-80 (1995) [6] Lamb, H.: Hydrodynamics. (1945) · Zbl 0828.01012 [7] Whitham, G. B.: Linear and non-linear waves. (1974) · Zbl 0373.76001 [8] Ovsyanniicov, L. V.; Makarenko, N. I.; Nalimov, V. I.: Non-linearproblems of the theory of surface and internal waves. (1985) [9] Zakharov, V. Ye: Stability of periodic waves of finite amplitude on the surface of a deep liquid. Zh. prikl. Mekh. tekh. Fiz. 2, 84-86 (1968) [10] Dobrokhotov, S. Yu: Non-local analogs of the Boussinesq non-linear equation for surface waves over uneven bottom and their asymptotic solutions. Dokl akad. Nauk SSSR 292, No. 1, 63-67 (1987) [11] Zakharov, V. Ye; Manakov, S. V.; Novikov, S. P.; Pitayevskii, L. P.: Soliton theory. The inverse problem method. (1980) [12] Newell, A. C.: Solitons in mathematics and physics. (1985) · Zbl 0565.35003 [13] Kodama, Y.: Normal forms for weakly dispersive wave equations. Phys. lett. Ser. A. 112, No. 5, 193-196 (1985) [14] Weidman, E. D.; Maxworthy, T.: Experiments on strong interactions between solitary waves. J. fluid mech. 85, No. 3, 417-431 (1978)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.