American Institute of Mathematical Sciences

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

Conformal and Geometric Properties of the Camassa-Holm Hierarchy

 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.
Citation: Rossen I. Ivanov. Conformal and Geometric Properties of the Camassa-Holm Hierarchy. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 545-554. doi: 10.3934/dcds.2007.19.545
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