Journal of Nonlinear Science

, Volume 20, Issue 3, pp 277–308 | Cite as

Computation of Time-Periodic Solutions of the Benjamin–Ono Equation

  • David M. AmbroseEmail author
  • Jon Wilkening
Open Access


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.


Periodic solutions Benjamin–Ono equation Nonlinear waves Solitons Bifurcation Continuation Optimal control Adjoint equation Spectral method 

Mathematics Subject Classification (2000)

65K10 37M20 35Q53 37G15 


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Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Department of Mathematical SciencesClemson UniversityClemsonUSA
  2. 2.Department of Mathematics and Lawrence Berkeley National LaboratoryUniversity of CaliforniaBerkeleyUSA

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