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$q$-Deformed KP Hierarchy and $q$-Deformed Constrained KP Hierarchy
Jingsong Heab,
Yinghua Lib,
Yi Chengb a Centre for Scientific Computing, University of Warwick, Coventry CV4 7AL, United Kingdom
b Department of Mathematics, University of Science and Technology of China, Hefei, 230026 Anhui, P.R. China
Abstract:
Using the determinant representation of gauge transformation operator, we have shown that the general form of
$\tau$ 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
$\tau$ 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
$x\to0$ and
$q\to1$, simultaneously.
Keywords:
$q$-deformation; $\tau$ function; Gauge transformation operator; $q$-KP hierarchy; $q$-cKP hierarchy.
MSC: 37K10;
35Q51;
35Q53;
35Q55 Received: January 27, 2006; in final form
April 28, 2006; Published online
June 13, 2006
Language: English
DOI:
10.3842/SIGMA.2006.060
Fulltext:
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