Journal article
Computation of Time-Periodic Solutions of the Benjamin-Ono Equation
Journal of nonlinear science, v 20(3), pp 277-308
01 Jun 2010
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Abstract
We present a spectrally accurate numerical method for finding nontrivial time-periodic solutions of nonlinear partial differential equations. The method is based on minimizing a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We solve an adjoint PDE to compute the gradient of this functional with respect to the initial condition. We include additional terms in the functional to specify the free parameters, which in the case of the Benjamin-Ono equation, are the mean, a spatial phase, a temporal phase, and the real part of one of the Fourier modes at t=0.
We use our method to study global paths of nontrivial time-periodic solutions connecting stationary and traveling waves of the Benjamin-Ono equation. As a starting guess for each path, we compute periodic solutions of the linearized problem by solving an infinite dimensional eigenvalue problem in closed form. We then use our numerical method to continue these solutions beyond the realm of linear theory until another traveling wave is reached. By experimentation with data fitting, we identify the analytical form of the solutions on the path connecting the one-hump stationary solution to the two-hump traveling wave. We then derive exact formulas for these solutions by explicitly solving the system of ODEs governing the evolution of solitons using the ansatz suggested by the numerical simulations.
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Details
- Title
- Computation of Time-Periodic Solutions of the Benjamin-Ono Equation
- Creators
- David M. Ambrose - Clemson UniversityJon Wilkening - University of California
- Publication Details
- Journal of nonlinear science, v 20(3), pp 277-308
- Publisher
- Springer Nature
- Number of pages
- 32
- Grant note
- DMS-0926378 / National Science Foundation; National Science Foundation (NSF) DE-AC02-05CH11231 / Office of Science, Computational and Technology Research, 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:000277298300001
- Scopus ID
- 2-s2.0-77955662134
- Other Identifier
- 991019173902704721
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- Collaboration types
- Domestic collaboration
- Web of Science research areas
- Mathematics, Applied
- Mechanics
- Physics, Mathematical