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September  2007, 19(3): 545-554. doi: 10.3934/dcds.2007.19.545

Conformal and Geometric Properties of the Camassa-Holm Hierarchy

Rossen I. Ivanov 1,

1. 

School of Mathematics, Trinity College Dublin, Dublin 2, Ireland

Received  March 2007 Revised  May 2007 Published  July 2007

Integrable equations with second order Lax pair like KdV and Camassa-Holm (CH) exhibit interesting conformal properties and can be written in terms of the so-called conformal invariants (Schwarz form). These properties for the CH hierarchy are discussed in this contribution.
    The squared eigenfunctions of the spectral problem, associated to the Camassa-Holm equation represent a complete basis of functions, which helps to describe the Inverse Scattering Transform (IST) for the Camassa-Holm hierarchy as a Generalised Fourier Transform (GFT). Using GFT we describe explicitly some members of the CH hierarchy, including integrable deformations for the CH equation. Also we show that solutions of some 2+1-dimensional generalizations of CH can be constructed via the IST for the CH hierarchy.
Keywords: Schwarz derivative, Virasoro algebra, Lax pair, conformal invariants, inverse scattering, solitons..
Mathematics Subject Classification: Primary: 37K10, 37K15; Secondary: 37K3.
Citation: Rossen I. Ivanov. Conformal and Geometric Properties of the Camassa-Holm Hierarchy. Discrete & Continuous Dynamical Systems, 2007, 19 (3) : 545-554. doi: 10.3934/dcds.2007.19.545
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