Solutions of a generalized constrained discrete KP hierarchy

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) 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.