Journal article
GLOBAL PATHS OF TIME-PERIODIC SOLUTIONS OF THE BENJAMIN-ONO EQUATION CONNECTING PAIRS OF TRAVELING WAVES
Communications in applied mathematics and computational science, v 4(1), pp 177-215
01 Jan 2009
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Abstract
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.
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Details
- Title
- GLOBAL PATHS OF TIME-PERIODIC SOLUTIONS OF THE BENJAMIN-ONO EQUATION CONNECTING PAIRS OF TRAVELING WAVES
- Creators
- David M. Ambrose - Drexel UniversityJon Wilkening - Lawrence Berkeley National Laboratory
- Publication Details
- Communications in applied mathematics and computational science, v 4(1), pp 177-215
- Publisher
- Mathematical Science Publ
- Number of pages
- 39
- Grant note
- DMS-0926378 / National Science Foundation; National Science Foundation (NSF) DE-AC02-05CH11231 / U.S. Department of Energy; United States Department of Energy (DOE)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000281881800007
- Scopus ID
- 2-s2.0-84992479394
- Other Identifier
- 991019168679204721
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- Collaboration types
- Domestic collaboration
- Web of Science research areas
- Mathematics, Applied
- Physics, Mathematical