Theoretical and Mathematical Physics

, Volume 137, Issue 1, pp 1459–1471 | Cite as

Nonintegrability of a Fifth-Order Equation with Integrable Two-Body Dynamics

  • D. D. Holm
  • A. N. W. Hone


We consider a fifth-order partial differential equation (PDE) that is a generalization of the integrable Camassa–Holm equation. This fifth-order PDE has exact solutions in terms of an arbitrary number of superposed pulsons with a geodesic Hamiltonian dynamics that is known to be integrable in the two-body case N = =2. Numerical simulations show that the pulsons are stable, dominate the initial value problem, and scatter elastically. These characteristics are reminiscent of solitons in integrable systems. But after demonstrating the nonexistence of a suitable Lagrangian or bi-Hamiltonian structure and obtaining negative results from Painlevé analysis and the Wahlquist–Estabrook method, we assert that this fifth-order PDE is not integrable.

Hamiltonian dynamics nonintegrability elastic scattering pulsons 


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Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • D. D. Holm
    • 1
  • A. N. W. Hone
    • 2
  1. 1.Theoretical Division and Center for Nonlinear StudiesLos Alamos National LaboratoryLos AlamosUSA
  2. 2.Institute of Mathematics and StatisticsUniversity of KentCanterburyUK

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