Conformal and Geometric Properties of the Camassa-Holm Hierarchy

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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

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

Received March 2007 Revised May 2007 Published July 2007

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

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