zbMATH — the first resource for mathematics

Formation of nonlinear wave conductors at resonance interaction of three surface waves. (English. Russian original) Zbl 0881.76014
J. Appl. Math. 61, No. 2, 183-193 (1997); translation from Prikl. Mat. Mekh. 61, No. 2, 190-201 (1997).
The authors investigate resonance interaction of three two-dimensional wave packets of capillary and gravity waves on the surface of infinite deep ideal fluid. The existence of solutions of wave conductor type is proved and their stability is investigated with respect to the slow changes of the wave conductor boundaries and amplitude of waves along the wave conductor.
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76E05 Parallel shear flows in hydrodynamic stability
[1] Kaup, D. J.: The three wave interaction–a nondispersive phenomenon. Stud. appl math. 55, No. 1, 9-44 (1976)
[2] Kaup, D. J.; Reiman, A.; Bers, A.: Space-time evolution of nonlinear three wave interaction. I. Rev. modern. Phys 51, No. 2, 275-309 (1979)
[3] Zakharov, V. Ye; Manakov, S. V.: The theory of resonant interaction of wave packets in non-linear media. Zh. eksp. Teor fiz. 69, No. 5, 1654-1673 (1975)
[4] Sukhorukov, A. E.: Non-linear wave interaction in optics and radio physics. (1988)
[5] Sakharov, V. Ye: The stability of periodic waves of finite amplitude on the surface of a deep liquid. Zh. prikl. Mekh. tekh. Fiz. 2, 86-94 (1968)
[6] Marchenko, A. V.: The Hamiltonian approach to investigating potential motions of an ideal liquid. Prikl. mat mekh. 59, No. 1, 102-108 (1995)
[7] Landau, L. D.; Lifshits, Ye.M: Theoretical physics, vol. 6. Hydrodynamics. 6 (1965)
[8] Landau, L. D.; Lifshits, Ye.M: Theoretical physics, vol. 7. Theory of elasticity. 7 (1965)
[9] Craik, A. D. D: Wave interaction and fluid flows. (1985) · Zbl 0581.76002
[10] Sibgatullin, N. R.: Lie-beklund groups of some model equations of the mechanics of a continuous medium and classical fields. Dokl. akad. Nauk SSSR. 291, No. 1, 302-305 (1986)
[11] Marchenko, A. V.; Sibgatullin, N. R.: The evolution of wave packets in three-wave interaction in a heavy liquid with a covering of ice. Izv. akad nauk SSSR. Mzhg. 6, 57-64 (1987) · Zbl 0657.76024
[12] Kantorovich, L. V.; Akilov, G. E.: Functional analysis. (1984) · Zbl 0555.46001
[13] Pliss, V. A.: The principle of reduction in the theory of the stability of motion. Izv. akad nauk SSSR ser. Mat. 28, No. 6, 1297-1324 (1964) · Zbl 0131.31505
[14] Iooss, G.; Adelmeyer, M.: Topics in bifurcation theory and applications. (1992) · Zbl 0833.34001
[15] Iooss, G.; Kirchgassner, K.: Water waves for small surface tension: an approach via normal form. Proc. roy. Soc. Edinburgh 122A, 267-299 (1992) · Zbl 0767.76004
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.