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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Additional info for A Bifurcation Theory for Three-Dimensional Oblique Travelling Gravity-Capillary Water Waves
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.  DUISTERMAAT, J. J. 1984. Bifurcation of periodic solutions near equilibrium points of Hamiltonian systems.
Craig and Nicholls  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  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  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.  MIELKE, A. 1988. Reduction of quasilinear elliptic equations in cylindrical domains with applications. Math. Meth. Appl. Sci. 10, 51–66.  MIELKE, A. 1991. Hamiltonian and Lagrangian Flows on Center Manifolds. Berlin: SpringerVerlag.  MOSER, J. 1976. Periodic orbits near an equilibrium and a theorem by Alan Weinstein. Commun. Pure Appl. Math. 29, 727–747.  PEGO, R. , AND QUINTERO, J. 2002. A host of traveling waves in a model of threedimensional water-wave dynamics.