Differential-difference evolution equations. II (Darboux transformation for the Toda lattice)

Darboux transformation and perturbation of linear functionals

“…The Darboux transformation was introduced in the classical monograph [6], although only the case of a (continuous) Sturm-Liouville operator is discussed there. Discrete versions were introduced later and their systematic study was undertaken by Matveev and Salle [19], who are responsible for the name "Darboux transformation".…”

“…In our work, we consider the general problem and we introduce in a natural way the so-called Darboux transformation in order to findJ 3 from J . The Darboux transformation is related with bispectral problems [11,12,14], as well as with differential evolution equations [19]. More recently, the analysis of rational spectral transformations, self-similarity, and orthogonal polynomials has been considered in [20,22].…”

Binary Darboux Transformation and Quasideterminant Solutions of The Chiral Field

“…The Darboux transformation [2,3] is one of the well-known methods of obtaining multisoliton solutions of many integrable systems [4][5][6][7][8]. We define the Darboux transformation on the matrix solutions to the Lax pair (1.5) and (1.6), in terms of an N × N matrix D(x + , x − , λ), called the Darboux matrix.…”

“…In this paper, we extend some of the earlier results by studying the binary Darboux transformation of the chiral field and obtain the quasideterminant solutions. Binary Darboux transformation is a special type of Darboux transformation, obtained by a combination of Darboux transformation in direct and adjont space ( [4][5][6][7][8][9][10][11][12][13][14][15][16]). The advantage of using binary Darboux transformation is that we obtain the grammian type solutions for the linear system and the potentails are also expressed in terms of quasideterminants.…”

A new algorithm for computing the Geronimus transformation with large shifts

“…This application is connected to the problem of classifying all sequences of orthogonal polynomials such that its derivatives form another set of orthogonal polynomials. In the last two decades, these transformations have attracted the interest of various specialists in different branches of mathematics and mathematical physics for their applications to different topics such as Discrete Integrable Systems [20,22,23], Quantum Mechanics, Bispectral Transformations in Orthogonal Polynomials [16][17][18], and Numerical Analysis [5,7,8,10,12].…”