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

Article, Preprint English OPEN
Jingsong He ; Yinghua Li ; Yi Cheng (2006)
  • Publisher: National Academy of Science of Ukraine
  • Journal: Symmetry (issn: 1815-0659)
  • Related identifiers: doi: 10.3842/SIGMA.2006.060
  • Subject: Mathematical Physics | $q$-KP hierarchy | Nonlinear Sciences - Exactly Solvable and Integrable Systems | Mathematics | Gauge transformation operator | $q$-deformation | QA1-939 | $au$ function | $q$-cKP hierarchy
    arxiv: Nonlinear Sciences::Exactly Solvable and Integrable Systems

Using the determinant representation of gauge transformation operator, we have shown that the general form of $au$ 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 $au$ 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 $x o 0$ and $q o 1$, simultaneously.
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