Tohoku Mathematical Journal
2011
June
SECOND SERIES VOL. 63, NO. 2

Tohoku Math. J.
63 (2011), 183-215

Title THE QUATERNIONIC KP HIERARCHY AND CONFORMALLY IMMERSED 2-TORI IN THE 4-SPHERE

Author Ian McIntosh

(Received April 12, 2010, revised December 17, 2010)
Abstract. The quaternionic KP hierarchy is the integrable hierarchy of p.d.e obtained by replacing the complex numbers with the quaternions in the standard construction of the KP hierarchy and its solutions; it is equivalent to what is often called the Davey-Stewartson II hierarchy. This article studies its relationship with the theory of conformally immersed tori in the 4-sphere via quaternionic holomorphic geometry. The Sato-Segal-Wilson construction of KP solutions is adapted to this setting and the connection with quaternionic holomorphic curves is made. We then compare three different notions of “spectral curve”: the QKP spectral curve; the Floquet multiplier spectral curve for the related Dirac operator; and the curve parameterising Darboux transforms of a conformal 2-torus in the 4-sphere.

2000 Mathematics Subject Classification. Primary 35Q53; Secondary 14H70, 53A30, 53C42.
Key words and phrases. Integrable systems, conformally immersed tori, quaternionic holomorphic curves, spectral curves.

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