The Darboux Matrices With a Single Multiple Pole and Their Applications

Darboux transformation (DT) plays a key role in constructing explicit closed-form solutions of completely integrable systems. This paper provides an algebraic construction of Darboux matrices with a single multiple pole for the $2\times 2$ Lax pair, in which the coefficient matrices are polynomials of spectral parameter. This special DT can handle the case where the spectral parameter coincides with its conjugate spectral parameter under non-Hermitian reduction. 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 DT in general form. That is, 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 DTs.