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

Abstract

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 onemore » 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.« less

Authors:
;
Publication Date:
Research Org.:
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Sponsoring Org.:
Computational Research Division
OSTI Identifier:
944566
Report Number(s):
LBNL-1298E
TRN: US200902%%863
DOE Contract Number:  
DE-AC02-05CH11231
Resource Type:
Journal Article
Journal Name:
Communication in Applied Mathematics and Computational Science
Additional Journal Information:
Journal Name: Communication in Applied Mathematics and Computational Science
Country of Publication:
United States
Language:
English
Subject:
97; AMPLITUDES; BIFURCATION; EXACT SOLUTIONS

Citation Formats

Ambrose, David M, and Wilkening, Jon. Global paths of time-periodic solutions of the Benjamin-Ono equation connecting arbitrary traveling waves. United States: N. p., 2008. Web.
Ambrose, David M, & Wilkening, Jon. Global paths of time-periodic solutions of the Benjamin-Ono equation connecting arbitrary traveling waves. United States.
Ambrose, David M, and Wilkening, Jon. 2008. "Global paths of time-periodic solutions of the Benjamin-Ono equation connecting arbitrary traveling waves". United States. https://www.osti.gov/servlets/purl/944566.
@article{osti_944566,
title = {Global paths of time-periodic solutions of the Benjamin-Ono equation connecting arbitrary traveling waves},
author = {Ambrose, David M and Wilkening, Jon},
abstractNote = {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.},
doi = {},
url = {https://www.osti.gov/biblio/944566}, journal = {Communication in Applied Mathematics and Computational Science},
number = ,
volume = ,
place = {United States},
year = {Thu Dec 11 00:00:00 EST 2008},
month = {Thu Dec 11 00:00:00 EST 2008}
}