Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 060, 32 pages      nlin.SI/0606039      http://dx.doi.org/10.3842/SIGMA.2006.060

q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy

Jingsong He a, b, Yinghua Li a and Yi Cheng a
a) Department of Mathematics, University of Science and Technology of China, Hefei, 230026 Anhui, P.R. China
b) Centre for Scientific Computing, University of Warwick, Coventry CV4 7AL, United Kingdom

Received January 27, 2006, in final form April 28, 2006; Published online June 13, 2006

Abstract
Using the determinant representation of gauge transformation operator, we have shown that the general form of τ function of the q-KP hierarchy is a q-deformed generalized Wronskian, which includes the q-deformed Wronskian as a special case. On the basis of these, we study the q-deformed constrained KP (q-cKP) hierarchy, i.e. l-constraints of q-KP hierarchy. Similar to the ordinary constrained KP (cKP) hierarchy, a large class of solutions of q-cKP hierarchy can be represented by q-deformed Wronskian determinant of functions satisfying a set of linear q-partial differential equations with constant coefficients. We obtained additional conditions for these functions imposed by the constraints. In particular, the effects of q-deformation (q-effects) in single q-soliton from the simplest τ function of the q-KP hierarchy and in multi-q-soliton from one-component q-cKP hierarchy, and their dependence of x and q, were also presented. Finally, we observe that q-soliton tends to the usual soliton of the KP equation when x 0 and q 1, simultaneously.

Key words: q-deformation; τ function; Gauge transformation operator; q-KP hierarchy; q-cKP hierarchy.

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