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By Groves M.D., Haragus M.

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

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F. 1996. A plethora of solitary gravity-capillary water waves with nearly critical Bond and Froude numbers. Phil. Trans. Roy. Soc. London A 354, 575–607. , AND NICHOLLS, D. P. 2000. Traveling two and three dimensional capillary gravity water waves. SIAM J. Math. 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.

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. [18] JONES, M. C. W. 1989. Small amplitude gravity-capillary waves in a channel of finite depth.

3 above: As ν is decreased through a critical value νc , four complex eigenvalues become purely imaginary by colliding in pairs on the imaginary axis, and we introduce a bifurcation parameter by writing ν = νc + µ. Observe that there are two degenerate versions of the Hamiltonian-Hopf bifurcation (both of which are included in the theory below), in which the eigenvalues are zero at criticality (see Figure 7); in one of these cases the eigenvalues are complex for µ > 0, while in the other (θ1 = ±π /2, θ2 = 0) they are real.

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A Bifurcation Theory for Three-Dimensional Oblique Travelling Gravity-Capillary Water Waves by Groves M.D., Haragus M.

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