The generalized Darboux matrices with the same poles and their applications

TL;DR

This paper develops an algebraic framework for constructing generalized Darboux matrices with the same poles for 2x2 Lax pairs, enabling explicit solutions and transformations for integrable systems like NLS and Kaup-Boussinesq equations.

Contribution

It introduces a classification of monic Darboux matrices with shared poles and provides explicit constructions and invariance properties, advancing solution methods for integrable systems.

Findings

Explicit construction of first-order monic Darboux matrices.

Derivation of n-th order Darboux matrices sharing the same pole.

Application to solutions of NLS and Kaup-Boussinesq equations.

Abstract

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.