Abstract:
Solutions of a generalized constrained discrete KP (gcdKP) hierarchy with the constraint $L^k=(L^k)_{\geq m}+\sum_{i=1}^lq_i\Delta^{-1}\Lambda^mr_i$ on the Lax operator are investigated by Darboux transformations $T_D(f)=f^{[1]}\cdot\Delta\cdot f^{-1}$ and $T_I(g)=(g^{[-1]})^{-1}\cdot\Delta^{-1}\cdot g$. Due to the special constraint on the Lax operator, it can be shown that the generating functions $f$ and $g$ of the corresponding Darboux transformations can only be chosen from {(}adjoint{\rm)} wave functions or $(L^k)_{<m}=\sum_{i=1}^lq_i\Delta^{-1}\Lambda^mr_i$. We discuss successive application of Darboux transformations for the gcdKP hierarchy. Solutions of the gcdKP hierarchy are obtained from $L^{\{0\}}=\Lambda$ by Darboux transformations, with a method that is highly nontrivial due to the special constraint on the Lax operator.
Keywords:discrete KP hierarchy, Darboux transformation, tau function.
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