Time-periodic solutions of the Benjamin-Ono equation Metadata

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Title

  • Main Title Time-periodic solutions of the Benjamin-Ono equation

Creator

  • Author: Ambrose , D.M.
    Creator Type: Personal
  • Author: Wilkening, Jon
    Creator Type: Personal

Contributor

  • Sponsor: Lawrence Berkeley National Laboratory. Computational Research Division.
    Contributor Type: Organization

Publisher

  • Name: Lawrence Berkeley National Laboratory
    Place of Publication: Berkeley, California
    Additional Info: Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley, CA (United States)

Date

  • Creation: 2008-04-01

Language

  • English

Description

  • Content Description: We present a spectrally accurate numerical method for finding non-trivial time-periodic solutions of non-linear 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 non-trivial 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 (or until the solution blows up). 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 ODE's governing the evolution of solitons using the ansatz suggested by the numerical simulations.

Subject

  • Keyword: Partial Differential Equations
  • Keyword: Solitons
  • Keyword: Functionals
  • Keyword: Eigenvalues
  • STI Subject Categories: 97

Source

  • Journal Name: Journal of Nonlinear Science

Collection

  • Name: Office of Scientific & Technical Information Technical Reports
    Code: OSTI

Institution

  • Name: UNT Libraries Government Documents Department
    Code: UNTGD

Resource Type

  • Article

Format

  • Text

Identifier

  • Report No.: LBNL-1306E
  • Grant Number: DE-AC02-05CH11231
  • Office of Scientific & Technical Information Report Number: 945045
  • Archival Resource Key: ark:/67531/metadc901687
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