Vol. 4, No. 1, 2009

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Global paths of time-periodic solutions of the Benjamin–Ono equation connecting pairs of traveling waves

David M. Ambrose and Jon Wilkening

Vol. 4 (2009), No. 1, 177–215

We classify all bifurcations from traveling waves to nontrivial 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 nontrivial solutions beyond the realm of linear theory. These paths are found to either reconnect 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 nontrivial solutions, but we do not attempt to analyze their analytic structure.

periodic solutions, Benjamin–Ono equation, nonlinear waves, solitons, bifurcation, continuation, exact solution, adjoint equation, spectral method
Mathematical Subject Classification 2000
Primary: 35Q53, 37G15, 37M20, 65K10
Received: 25 November 2008
Revised: 12 July 2009
Accepted: 21 July 2009
Published: 25 November 2009
David M. Ambrose
Department of Mathematics
Drexel University
Philadelphia, PA 19104
United States
Jon Wilkening
Department of Mathematics and Lawrence Berkeley National Laboratory
University of California
Berkeley, CA 94720
United States