Laws of composition of Bäcklund transformations and the universal form of completely integrable systems in dimensions two and three

Abstract

Bacldund transformations are defined as operations on solutions of a Riemann boundary value problem (vector bundles over P1) that add apparent singularities. For solutions of difference and differential linear spectral problems, BAcldund transformations are presented in explicit form through the Christoffel formula and its generalizations. Identities satisfied by iterations of elementary BAcldund transformations are represented in the form of the law of addition or as the three-dimensional difference equation of Hirota's type. Matrix two-dimensional isospectral deformation equations are imbedded into three-dimensional scalar systems of Kadomtzev-Petviashvili (law of addition) form. Two-dimensional matrix systems correspond to reductions of Kadomtzev-Petviashvili equations with pseudodifferential operators satisfying algebraic equations.The purpose of this article is to demonstrate how various completely integrable systems in dimensions two and three can be represented in a transparent form, in terms of identities implied by BAcklund transformations (BTs) algebra. Investigations of BTs (1-3) have shown how a successive application of BTs replaces continuous variables by discrete ones. In this way one constructs, starting from a two-dimensional completely integrable system (say, the operator nonlinear Schrodinger equation), lattice systems in one or two dimensions equivalent to and approximating the initial one. Here we classify lattice models through identities satisfied by BTs. In various infinitesimal limits and reductions, one recovers a variety of two-and three-dimensional difference, difference-differential, and differential, completely integrable systems of isospectral deformation origin. In this there is an analogy with Hirota's bilinear method (4). (7) and its generalizations. Christoffel's formula for the iteration of BTs corresponds to the equivalent situation when the weight function w(A) is multiplied by a function rational in A and a finite sum of 8 functions is added; see treatments by Szego (6) and Uvarov (8). In Uvarov's formula (8) In analogy to Eq. 2, if poles instead of zeros are added tof(A), then in Eq. 3 the corresponding polynomials of the first kind are substituted by polynomials of the second kind.Alternatively, the effect of an elementary BT can be described in terms of a single difference operator X(A). This operator X(A) is a difference analog of the vertex operator that represents the generating function for infinite dimensional affine algebras described by Lepowsky and Wilson (9). An operator X(A) acts on functions of infinitely many variables {ai} that are

About several classes of bi-orthogonal polynomials and discrete integrable systems

“…With the help of (4) and (5), we can also obtain a Lax pair for (6) and (7). Let Ψ l n = (P l+m n , P l+m−1 n , · · · , P l n ) T ,…”

“…Emails: changxk@lsec.cc.ac.cn,chenxm@lsec.cc.ac.cn,hxb@lsec.cc.ac.cn,tam@comp.hkbu.edu.hk There has been growing interest in studying the relationship between discrete integrable systems and orthogonal polynomials, as orthogonal polynomials provide a powerful tool for studying discrete integrable systems [7,21]. For example, in [17], the quotient-difference (QD) algorithm [18] e l+1 k q l+1 k = e l k q l k+1 , e l+1 k + q l+1 k+1 = e l k+1 + q l k+1 (1) is nothing but the compatibility condition of the spectral problem related to the discrete-time Toda equation [11]…”

Direct and inverse polynomial perturbations of hermitian linear functionals

“…However, its importance goes much further, among other things, due to the connection with such an outstanding idea as the Darboux transformation (see [4,[21][22][23]35,43]). This connection has led to unexpectedly fruitful relations with other subjects like, for instance, spectral theory, integrable systems or quantum physics (see for instance [7,30,32,[36][37][38]). It is worth noting that the interest of the Darboux transformation goes beyond the study of the discrete systems underlying the difference equations related to orthogonal polynomials.…”

Discrete Darboux transformations, the discrete-time Toda lattice, and the Askey–Wilson polynomials

“…There are many common links between Darboux transformations [2, 10, 16, 18, 20-24, 27, 33, 34, 44, 48, 50, 53-55, 62, 63, 81-83, 85, 86] and symmetry properties of the classical special functions [25,75]. Darboux transformations are very much related to the Backlund and dressing transformations of the theory of solitons [20,24,50,53,54,76,81,85,86]. Firstly, they form a core of the factorization method [2,10,23,44,55,62,63,68]; secondly, in the theory of orthogonal polynomials their analogs go back to Christoffel [16,18,20,27,48,75,82]; and thirdly, in numerical calculations of matrix eigenvalues, they appear in the procedure called the LiJ-algorithm [27,83] (a modified form of which is known also as the QR-algorithm).…”

“…Darboux transformations are very much related to the Backlund and dressing transformations of the theory of solitons [20,24,50,53,54,76,81,85,86]. Firstly, they form a core of the factorization method [2,10,23,44,55,62,63,68]; secondly, in the theory of orthogonal polynomials their analogs go back to Christoffel [16,18,20,27,48,75,82]; and thirdly, in numerical calculations of matrix eigenvalues, they appear in the procedure called the LiJ-algorithm [27,83] (a modified form of which is known also as the QR-algorithm). As in all these cases the underlying transformations share many common properties, we refer to them as Darboux transformations and add the adjective "discrete" in the context of finitedifference equations.…”