By Groves M.D., Haragus M.

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

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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.

Craig and Nicholls [5] have recently given results of this kind using a variational Lyapunov-Schmidt reduction of the equations for travelling water waves. In particular, both approaches give existence theories for the “short-crested waves” described by Hammack, Scheffner, and Segur [12] which are spatially periodic in two orthogonal directions, one of which coincides with the direction of propagation (take θ1 = 0, θ2 = ±π /2) and for the doubly periodic waves described by Reeder and Shinbrot [29] whose fundamental domain is a “symmetric diamond” (choose θ1 = ±θ2 , κ = ν) (see Figure 10b).

A variational principle for a fluid with a free surface. J. Fluid Mech. 27, 395–397. [24] MIELKE, A. 1988. Reduction of quasilinear elliptic equations in cylindrical domains with applications. Math. Meth. Appl. Sci. 10, 51–66. [25] MIELKE, A. 1991. Hamiltonian and Lagrangian Flows on Center Manifolds. Berlin: SpringerVerlag. [26] MOSER, J. 1976. Periodic orbits near an equilibrium and a theorem by Alan Weinstein. Commun. Pure Appl. Math. 29, 727–747. [27] PEGO, R. , AND QUINTERO, J. 2002. A host of traveling waves in a model of threedimensional water-wave dynamics.

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