A Bifurcation Theory for Three-Dimensional Oblique by Groves M.D., Haragus M.

By Groves M.D., Haragus M.

This text provides a rigorous lifestyles concept for small-amplitude threedimensional vacationing water waves. The hydrodynamic challenge is formulated as an infinite-dimensional Hamiltonian approach within which an arbitrary horizontal spatial course is the timelike variable. Wave motions which are periodic in a moment, assorted horizontal path are detected utilizing a centre-manifold relief strategy in which the matter is decreased to a in the neighborhood similar Hamiltonian procedure with a finite variety of levels of freedom.

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Spatial wave dynamics of steady oblique wave interactions. Physica D 145, 207–232. , AND KIRCHGASSNER , K. 1990. Bifurcation d’ondes solitaires en pr´esence d’une faible tension superficielle. C. R. Acad. Sci. Paris, S´er. 1 311, 265–268. , AND KIRCHGASSNER , K. 1992. Water waves for small surface tension: An approach via normal form. Proc. Roy. Soc. Edinburgh A 122, 267–299. , AND PE´ ROUE` ME, M. C. 1993. Perturbed homoclinic solutions in reversible 1:1 resonance vector fields. J. Diff. Eq. 102, 62–88.

Anal. 32, 323–359. -C. 1997. Solitary waves of generalized Kadomtsev-Petviashvili equations. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 14, 211–236. , AND IOOSS, G. 2003. Water-waves as a spatial dynamical system. In Handbook of Mathematical Fluid Dynamics, vol. 2 (ed. Friedlander, S. ), pages 443–499. Amsterdam: North-Holland. , AND KHARIF, C. 1999. Nonlinear gravity and capillary-gravity waves. Ann. Rev. Fluid Mech. 31, 301–346. [9] DUISTERMAAT, J. J. 1984. Bifurcation of periodic solutions near equilibrium points of Hamiltonian systems.

The following result is obtained by applying the Weinstein-Moser theorem to the reduced Hamiltonian system and to its further reduction by the symmetry S2 ; in the latter case we recover the result given by Groves [10, Theorem 5] with the nonresonance condition removed. Theorem 5. Suppose that θ1 = ±π /2 and θ2 = 0. (i) Suppose that (β0 , α0 ) lies below the line C1 , to the left of C j and to the right of C j+1 for some j ∈ N. The reduced equations on the centre manifold have 2 j geometrically distinct periodic orbits on the energy surface { H˜ 0 = } for each sufficiently small value of > 0, and j of these orbits are invariant under S2 .

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