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

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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 propose a conjecture that gives the mapping from one bifurcation to its counterpart on the other ... continued below

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Ambrose, David M. & Wilkening, Jon December 11, 2008.

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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 propose a conjecture that gives the mapping from one bifurcation to its counterpart on the other side of the path of non-trivial solutions. By experimentation with data fitting, we identify the form of the exact solutions on the path connecting two traveling waves, which represents the Fourier coefficients of the solution as power sums of a finite number of particle positions whose elementary symmetric functions execute simple orbits in the complex plane (circles or epicycles). We then solve a system of algebraic equations to express the unknown constants in the new representation in terms of the mean, a spatial phase, a temporal phase, four integers (enumerating the bifurcation at each end of the path) and one additional bifurcation parameter. We also find examples of interior bifurcations from these paths of already non-trivial solutions, but we do not attempt to analyze their algebraic structure.

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  • Journal Name: Communication in Applied Mathematics and Computational Science

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  • Report No.: LBNL-1298E
  • Grant Number: DE-AC02-05CH11231
  • Office of Scientific & Technical Information Report Number: 944566
  • Archival Resource Key: ark:/67531/metadc896834

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  • December 11, 2008

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  • Sept. 27, 2016, 1:39 a.m.

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  • Nov. 8, 2016, 1:12 p.m.

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Ambrose, David M. & Wilkening, Jon. Global paths of time-periodic solutions of the Benjamin-Ono equation connecting arbitrary traveling waves, article, December 11, 2008; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc896834/: accessed September 24, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.