Solutions of generalized constrained discrete KP hierarchy

TL;DR

This paper investigates solutions to a generalized constrained discrete KP hierarchy using Darboux transformations, revealing specific constraints on generating functions and constructing explicit solutions from initial Lax operators.

Contribution

It introduces a novel approach to solving the gcdKP hierarchy by applying Darboux transformations under particular constraints on the Lax operator.

Findings

Solutions constructed from initial Lax operator using Darboux transformations.

Generating functions are restricted to wave functions or specific operator sums.

Successive Darboux transformations yield explicit solutions to the gcdKP hierarchy.

Abstract

Solutions of a generalized constrained discrete KP (gcdKP) hierarchy with constraint on Lax operator $L^k=(L^k)_{\geq m}+\sum_{i=1}^lq_i\Delta^{-1}\Lambda^mr_i$, are invesitigated 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 this special constraint on Lax operator, it is showed that the generating functions $f$ and $g$ of the corresponding Darboux transformations, can only be chosen from (adjoint) wave functions or $(L^k)_{<m}=\sum_{i=1}^lq_i\Delta^{-1}\Lambda^mr_i$. Then successive applications of Darboux transformations for gcdKP hierarchy are discussed. Finally based upon above, solutions of gcdKP hierarchy are obtained from $L^{\{0\}}=\Lambda$ by Darboux transformations.