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SUMMARY:Inverse scattering for the Camassa-Holm hierarchy - Rossen Ivanov 
 (Trinity College Dublin and Dublin Institute of Technology)
DTSTART:20081016T140000Z
DTEND:20081016T150000Z
UID:TALK14306@talks.cam.ac.uk
CONTACT:6743
DESCRIPTION:The inverse scattering method is developed for the Camassa–H
 olm equation. As an illustration  the solutions corresponding to the refle
 ctionless\npotentials are constructed in terms of the scattering data. The
  main difference with respect to the standard inverse scattering transform
  lies in the fact that we have a weighted spectral problem. This requires 
 different asymptotic expansions for the eigenfunctions.\n\nThe squared eig
 enfunctions of the spectral problem associated with the\nCamassa–Holm eq
 uation represent a complete basis of functions\, which helps\nto describe 
 the inverse scattering transform for the Camassa–Holm hierarchy as a gen
 eralized Fourier transform. There exists a completeness relation for the s
 quared solutions of the Camassa–Holm\nspectral problem. We show that all
  the fundamental properties of the Camassa–Holm equation such as the int
 egrals of motion\, the description of the equations of the whole hierarchy
  and their Hamiltonian structures can be naturally expressed making use of
  the completeness relation and the recursion operator\, whose eigenfunctio
 ns are the squared solutions.
LOCATION:MR14\, CMS
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