Using the determinant representation of gauge transformation operator, wehave shown that the general form of $\tau$ function of the $q$-KP hierarchy isa q-deformed generalized Wronskian, which includes the q-deformed Wronskian asa 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 theordinary constrained KP (cKP) hierarchy, a large class of solutions of $q$-cKPhierarchy can be represented by q-deformed Wronskian determinant of functionssatisfying a set of linear $q$-partial differential equations with constantcoefficients. We obtained additional conditions for these functions imposed bythe constraints. In particular, the effects of $q$-deformation ($q$-effects) insingle $q$-soliton from the simplest $\tau$ function of the $q$-KP hierarchyand in multi-$q$-soliton from one-component $q$-cKP hierarchy, and theirdependence 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.
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