By Groves M.D., Haragus M.
This text offers a rigorous lifestyles thought for small-amplitude threedimensional traveling water waves. The hydrodynamic challenge is formulated as an infinite-dimensional Hamiltonian procedure within which an arbitrary horizontal spatial path is the timelike variable. Wave motions which are periodic in a moment, varied horizontal course are detected utilizing a centre-manifold relief method wherein the matter is diminished to a in the neighborhood identical Hamiltonian process with a finite variety of levels of freedom.
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Additional resources for A Bifurcation Theory for Three-Dimensional Oblique Travelling Gravity-Capillary Water Waves
The 1 : −3 resonance at points of R3 . The coefficients in the formula (47) for the Hamiltonian in Birkhoff normal form are given by c1 = −q 3 (1 + 2 cosh 2q) csc2 θ2 csch3 q sech3 q ×(109634 + 192616 cosh 2q + 121897 cosh 4q + 55780 cosh 6q + 15022 cosh 8q + 2484 cosh 10q + 231 cosh 12q) ×[768(−4q − 8q cosh 2q + 4 sinh 2q + sinh 4q)2 ]−1 , c2 = 3q 3 csc2 θ2 csch3 q sech3 q ×(369855 + 667539 cosh 2q + 493080 cosh 4q + 282937 cosh 6q + 125052 cosh 8q + 41595 cosh 10q + 9320 cosh 12q + 1209 cosh 14q + 69 cosh 16q) ×[256(4 + 3 cosh 2q)(4q − 5 cosh q sinh q + 3 coth 3q sinh2 q)]−1 ×[12q − 3 sinh 2q − 3 sinh 4q + sinh 6q]−1 , c3 = 27q 3 (1 + 2 cosh 2q) csc2 θ2 csch3 q sech3 q ×(7381 + 1671 cosh 2q + 2373 cosh 4q + 2819 cosh 6q + 114 cosh 8q + 2409 cosh 10q + 1291 cosh 12q + 240 cosh 14q + 105 cosh 16q + 29 cosh 18q) ×[256(11 + 10 cosh 2q)(12q − 3 sinh 2q − 3 sinh 4q + sinh 6q)2 ]−1 .
1915. On ripples. Phil. Mag. 29, 688–700.
The Schmidt-Duistermaat theory is an existence theory for small-amplitude periodic solutions of two-degree-offreedom Hamiltonian systems with two pairs of imaginary eigenvalues in 1 : k or 1 : −k resonance, and the resonant eigenvalues ±iq, ±ikq are also allowed to depend upon a detuning parameter µ which takes them out of resonance for µ = 0. The theory is based upon a Lyapunov-Schmidt reduction (Schmidt) and singularity theory (Duistermaat), and is in essence a persistence result for periodic solutions of the fourth-order Birkhoff normal form associated with the resonance.
A Bifurcation Theory for Three-Dimensional Oblique Travelling Gravity-Capillary Water Waves by Groves M.D., Haragus M.