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

Jingsong He ; Yinghua Li ; Yi Cheng

Symmetry, integrability and geometry: methods and applications, Tome 2 (2006)

Résumé

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.

Détail

Keywords: $q$-deformation; $\tau$ function; Gauge transformation operator; $q$-KP hierarchy; $q$-cKP hierarchy.

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