Algebraic construction of the Darboux matrix revisited

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-Bäcklund transformation, based on different λ-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.

Properties of the temporal–spatial interference pattern during soliton interaction

“…For a BEC system, it is related to scattering length, and in nonlinear optics it is related to the nonlinear Kerr effect. The interaction between solitons can be studied based on multi-soliton solution which can be obtained by Bácklund transformation [25][26][27]. Here, for simplicity and without loss of generality, we will discuss two solitons' interaction based on the twosoliton solution.…”

A noncommutative coupled dispersionless system, Darboux transformation and explicit solutions

“…[15][16][17][18][19][20] The DT for the generalized CD system has been presented in Refs. 16 and 17. In this section, we present DT for the U (2) nc-CD system.…”

Darboux transformation and novel solutions for the long wave-short wave model

“…In paper [5], the author establishes the relations between different construction methods. However, the classical Darboux transformation can not tackle with the high-order spectral.…”