We present the integration of the “pair” flows associated to the Camassa–Holm (CH) hierarchy i.e., an explicit exact formula for the update of the initial velocity profile in terms of initial data when run by the flow associated to a Hamiltonian which (up to a constant factor) is given by the sum of the reciprocals of the squares of any two eigenvalues of the underlying spectral problem. The method stems from the integration of “individual” flows of the CH hierarchy described in [Loubet 2006; McKean 2003], and is seen to be more general in scope in that it may be applied when considering more complex flows (e.g., when the Hamiltonian involves an arbitrary number of eigenvalues of the associated spectral problem) up to when envisaging the full CH flow itself which is nothing but a superposition of commuting individual actions. Indeed, by incorporating piece by piece into the Hamiltonian the distinct eigenvalues describing the spectrum associated to the initial profile, we may recover McKean’s Fredholm determinant formulas [McKean 2003] expressing the evolution of initial data when acted upon by the full CH flow. We also give account of the large-time (and limiting remote past and future) asymptotics and obtain (partial) confirmation of the thesis about soliton genesis and soliton interaction raised in [Loubet 2006].

Loubet, E (2007). *Integration of pair flows of the Camassa-Holm hierarchy.* In: Pinsky, M; Birnir, B. Probability, geometry and integrable systems. Cambridge: Cambridge Univ. Press, 261-285.

## Abstract

We present the integration of the “pair” flows associated to the Camassa–Holm (CH) hierarchy i.e., an explicit exact formula for the update of the initial velocity profile in terms of initial data when run by the flow associated to a Hamiltonian which (up to a constant factor) is given by the sum of the reciprocals of the squares of any two eigenvalues of the underlying spectral problem. The method stems from the integration of “individual” flows of the CH hierarchy described in [Loubet 2006; McKean 2003], and is seen to be more general in scope in that it may be applied when considering more complex flows (e.g., when the Hamiltonian involves an arbitrary number of eigenvalues of the associated spectral problem) up to when envisaging the full CH flow itself which is nothing but a superposition of commuting individual actions. Indeed, by incorporating piece by piece into the Hamiltonian the distinct eigenvalues describing the spectrum associated to the initial profile, we may recover McKean’s Fredholm determinant formulas [McKean 2003] expressing the evolution of initial data when acted upon by the full CH flow. We also give account of the large-time (and limiting remote past and future) asymptotics and obtain (partial) confirmation of the thesis about soliton genesis and soliton interaction raised in [Loubet 2006].

## Citations

## Altmetrics

## Downloads

## Additional indexing

Item Type: | Book Section, refereed, original work |
---|---|

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Uncontrolled Keywords: | integrable systems, soliton traveling waves, spectral theory, Darboux transformations, asymptotic analysis. |

Language: | English |

Date: | 2007 |

Deposited On: | 09 Nov 2009 01:20 |

Last Modified: | 05 Apr 2016 13:23 |

Publisher: | Cambridge Univ. Press |

Series Name: | Mathematical Sciences Research Institute publications |

Number: | 55 |

ISBN: | 978-0-521-89527-9 |

Official URL: | http://www.msri.org/communications/books/Book55/files/12loubet.pdf |

Related URLs: | http://opac.nebis.ch:80/F/?local_base=NEBIS&con_lng=GER&func=find-b&find_code=SYS&request=005593146 http://www.msri.org/communications/books/Book55/index.html |

## Download

TrendTerms displays relevant terms of the abstract of this publication and related documents on a map. The terms and their relations were extracted from ZORA using word statistics. Their timelines are taken from ZORA as well. The bubble size of a term is proportional to the number of documents where the term occurs. Red, orange, yellow and green colors are used for terms that occur in the current document; red indicates high interlinkedness of a term with other terms, orange, yellow and green decreasing interlinkedness. Blue is used for terms that have a relation with the terms in this document, but occur in other documents.

You can navigate and zoom the map. Mouse-hovering a term displays its timeline, clicking it yields the associated documents.