Henrik Aratyn, Jose Francisco Gomes, Abraham H. Zimerman, “On a Negative Flow of the AKNS Hierarchy and Its Relation to a Two-Component Camassa–Holm Equation”, SIGMA, 2006, Volume 2,070, 12 pp.

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On a Negative Flow of the AKNS Hierarchy and Its Relation to a Two-Component Camassa–Holm Equation

Henrik Aratyn^a, Jose Francisco Gomes^b, Abraham H. Zimerman^b

^a Department of Physics, University of Illinois at Chicago, 845 W. Taylor St., Chicago, Illinois 60607-7059
^b Instituto de Física Teórica-UNESP, Rua Pamplona 145, 01405-900 São Paulo, Brazil

Abstract: Different gauge copies of the Ablowitz–Kaup–Newell–Segur (AKNS) model labeled by an angle $\theta$ are constructed and then reduced to the two-component Camassa–Holm model. Only three different independent classes of reductions are encountered corresponding to the angle $\theta$ being 0, $\pi/2$ or taking any value in the interval $0<\theta<\pi/2$. This construction induces Bäcklund transformations between solutions of the two-component Camassa–Holm model associated with different classes of reduction.

Keywords: integrable hierarchies; Camassa–Holm equation; Bäcklund transformation.

MSC: 37K10; 35Q53; 53A07; 53B50

Received: September 13, 2006; in final form October 5, 2006; Published online October 17, 2006

Language: English

DOI: 10.3842/SIGMA.2006.070