Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2008-11-26
Comm. App. Math. and Comp. Sci., 4/1 (2009), pp. 177-215
Nonlinear Sciences
Exactly Solvable and Integrable Systems
35 pages, 14 figures; changed title slightly, added 7 references, changed conjecture to a theorem and proved it, moved some ma
Scientific paper
We classify all bifurcations from traveling waves to non-trivial time-periodic solutions of the Benjamin-Ono equation that are predicted by linearization. We use a spectrally accurate numerical continuation method to study several paths of non-trivial solutions beyond the realm of linear theory. These paths are found to either re-connect with a different traveling wave or to blow up. In the latter case, as the bifurcation parameter approaches a critical value, the amplitude of the initial condition grows without bound and the period approaches zero. We then prove a theorem that gives the mapping from one bifurcation to its counterpart on the other side of the path and exhibits exact formulas for the time-periodic solutions on this path. The Fourier coefficients of these solutions are power sums of a finite number of particle positions whose elementary symmetric functions execute simple orbits (circles or epicycles) in the unit disk of the complex plane. We also find examples of interior bifurcations from these paths of already non-trivial solutions, but we do not attempt to analyze their analytic structure.
Ambrose David M.
Wilkening Jon
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