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

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 $x\to 0$ and $q\to 1$, simultaneously.

Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

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Cited by in corpus (9)

• Kadomtsev-Petviashvili equation: one-constraint method and lump pattern
• The gauge transformation of the constrained semi-discrete KP hierarchy
• The compatibility of additional symmetry and gauge transformations for the constrained discrete Kadomtsev-Petviashvili hierarchy
• Sato theory on the $q$-Toda hierarchy and its extension
• The wronskian solution of the constrained discrete KP hierarchy
• Quasideterminant solutions to a noncommutative $q$-difference two-dimensional Toda lattice equation
• Integrability on generalized $q$-Toda equation and hierarchy
• The applications of the gauge transformation for the BKP hierarchy
• The integral type gauge transformation and the additional symmetry for the constrained KP hierarchy