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SIGMA 2 (2006), 060, 32 pages nlin.SI/0606039
https://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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