The generalized Darboux matrices with the same poles and their applications

Darboux transformation plays a key role in constructing explicit closed-form solutions of completely integrable systems. This paper provides an algebraic construction of generalized Darboux matrices with the same poles for the $2\times2$ Lax pair, in which the coefficient matrices are polynomials of spectral parameter. The first-order monic Darboux matrix is constructed explicitly and its classification theorem is presented. Then by using the solutions of the corresponding adjoint Lax pair, the $n$-order monic Darboux matrix and its inverse, both sharing the same unique pole, are derived explicitly. Further, a theorem is proposed to describe the invariance of Darboux matrix regarding pole distributions in Darboux matrix and its inverse. Finally, a unified theorem is offered to construct formal Darboux transformation in general form. All Darboux matrices expressible as the product of $n$ first-order monic Darboux matrices can be constructed in this way. The nonlocal focusing NLS equation, the focusing NLS equation and the Kaup-Boussinesq equation are taken as examples to illustrate the application of these Darboux transformations.