Algebraic construction of the Darboux matrix revisited

TL;DR

This paper reviews algebraic methods for constructing Darboux matrices in 1+1-dimensional integrable systems, emphasizing nonisospectral cases, and introduces new formulas for N-soliton solutions and constraints on Lax pairs.

Contribution

It provides a comprehensive review of Darboux matrix constructions with new formulas for N-soliton solutions and constraints on nonisospectral Lax pairs.

Findings

Derived symmetric N-soliton formulas for general cases

Presented linear and bilinear constraints on Lax pair matrices

Included new formulas for N-soliton surfaces

Abstract

We present algebraic construction of Darboux matrices for 1+1-dimensional integrable systems of nonlinear partial differential equations with a special stress on the nonisospectral case. We discuss different approaches to the Darboux-Backlund transformation, based on different lambda-dependencies of the Darboux matrix: polynomial, sum of partial fractions, or the transfer matrix form. We derive symmetric N-soliton formulas in the general case. The matrix spectral parameter and dressing actions in loop groups are also discussed. We describe reductions to twisted loop groups, unitary reductions, the matrix Lax pair for the KdV equation and reductions of chiral models (harmonic maps) to SU(n) and to Grassmann spaces. We show that in the KdV case the nilpotent Darboux matrix generates the binary Darboux transformation. The paper is intended as a review of known results (usually presented in a novel context) but some new results are included as well, e.g., general compact formulas for N-soliton surfaces and linear and bilinear constraints on the nonisospectral Lax pair matrices which are preserved by Darboux transformations.