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

Technique

By Groves M.D., Haragus M.

This text offers a rigorous lifestyles concept for small-amplitude threedimensional traveling 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, various horizontal path are detected utilizing a centre-manifold aid process through which the matter is lowered to a in the neighborhood identical Hamiltonian process with a finite variety of levels of freedom.

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Extra resources for A Bifurcation Theory for Three-Dimensional Oblique Travelling Gravity-Capillary Water Waves

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In the further special case θ1 = ±π/2, θ2 = 0, the mode n and mode −n eigenvalues for n = 0 again coincide to form geometrically double eigenvalues; this case was investigated in detail by Groves [10] and HaragusCourcelle and Kirchg¨assner [13]. 4. 1. Coordinates on the Centre Manifold We begin by choosing a symplectic basis for X1 consisting of generalised eigenvectors of L. The operator always has a zero eigenvalue with eigenvector v0 = (0, 0, 1, 0)T . Direct calculations show that this eigenvector has a Jordan chain of length 2 if α0 = sin2 θ2 , of 428 M.

Diff. Eq. 45, 113–127. Three-Dimensional Oblique Travelling Gravity-Capillary Water Waves 447 ¨ , K. 1988. Nonlinear resonant surface waves and homoclinic bifurcation. Adv. [20] KIRCHGASSNER Appl. Math. 26, 135–181. [21] LOMBARDI, E. 1997. Orbits homoclinic to exponentially small periodic orbits for a class of reversible systems. Application to water waves. Arch. Ratl. Mech. Anal. 137, 227–304. [22] LOMBARDI, E. 2000. Oscillatory Integrals and Phenomena beyond All Algebraic Orders. Berlin: Springer-Verlag.

20, 47–57. [33] WILTON, J. R. 1915. On ripples. Phil. Mag. 29, 688–700.

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