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.
Deep Dive into Algebraic construction of the Darboux matrix revisited.
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-sol
A 1+1-dimensional integrable system can be considered as integrability conditions for a linear problem (a system of linear partial differential equations defined by two matrices containing the spectral parameter), see for instance [56]. The Darboux-Bäcklund transform is a gauge-like transformation (defined by the Darboux matrix) which preserves the form of the linear problem [14,22,27,40,66]. All approaches to the construction of Darboux matrices originate in the dressing method [56,68,81,82].
The paper is intended as a presentation of Darboux-Bäcklund transformations from a unified perspective, first presented in [13,14]. The construction of the Darboux matrix is divided into two stages. First, we uniquely characterize the considered linear problem in terms of algebraic constraints (the divisor of poles, loop group reductions and other algebraic properties, e.g., linear and biblinear constraints). Then, we construct the Darboux matrix preserving all these constraints. Using general theorems, including those from the present paper, one may construct the Darboux matrix in a way which is almost algorithmic.
The paper is intended as a review of known results but some new results are also included. We discuss in detail elementary Darboux transformation (Darboux matrix which has a single simple zero), symmetric formulas for Darboux matrices and soliton surfaces (in the general case), and loop group reductions for polynomial Darboux matrices. Two examples are discussed in detail: the Korteweg-de Vries equation and chiral models (harmonic maps).
The part which seems to be most original contains the description of linear and bilinear invariants of Darboux transformations. We prove that multilinear constraints introduced in [14] are invariant with respect to the polynomial Darboux transformation (also in the nonisospectral case). Taking them into account we can avoid some cumbersome calculations, our construction assumes a more elegant form and, last but not least, we do not need any assumptions concerning boundary conditions.
Another important aim of this paper is to show similarities and even an equivalence between different algebraic approaches to the construction of the Darboux matrix. This is a novelty in itself because sometimes it is difficult to notice connections between different methods. The existing monographs, even the recent ones, focus on a chosen single approach, compare [24,27,46,47,56,60].
We consider a nonlinear system of partial differential equations which is equivalent to the compatibility conditions
for the following system of linear equations (known as the Lax pair, at least in the case of two independent variables)
where n × n matrices U ν depend on x 1 , . . . , x m and on the so called spectral parameter λ (and, as usual, Ψ, ν = ∂Ψ/∂x ν , etc.). We assume that Ψ is also a matrix (the fundamental solution of the linear system (1.2)). We fix our attention on the case m = 2 (although most results hold for any m) and shortly denote by x the set of all variables, i.e., x = (x 1 , . . . , x m ).
Let us recall that the most important characteristic of the matrices U 1 , U 2 is their dependence on the spectral parameter λ. In the typical case U ν are rational with respect to λ. Actually we will consider a more general situation. We assume that the Lax pair is rational with respect to λ, and
• “isospectral case”: λ is a constant parameter,
• “non-isospectral” case:
λ, ν = L ν (x, λ) , (ν = 1, . . . , m) ,
where L ν are given functions, rational with respect to λ (this case reduces to the isospectral one for L ν (x, λ) ≡ 0).
Remark 1.1. The differential equations (1.3) are of the first order, so their solution λ = Λ(x, ζ) depends on a constant of integration ζ which plays the role of the constant spectral parameter.
The solution of the system (1.3) exists provided that compatibility conditions hold, for more details see [14]. In general Λ = Λ(x, ζ) is an implicit function, although in many special cases explicit expression for Λ can be found, compare [11,14,69]).
The application of the dressing method to generate new solutions of nonlinear equations “coded” in (1.1) consists in the following (see [56,79,82]). Suppose that we are able to construct a gauge-like transformation Ψ = DΨ (where D = D(x, λ) will be called the Darboux matrix) such that the structure of matrices Ũν ,
is identical with the structure of the matrices U ν . The soliton fields entering U ν are replaced by some new fields which, obviously, have to satisfy the nonlinear system (1.1) as well.
Remark 1.2. The Darboux transformation should preserve divisors of poles (i.e., poles and their multiplicities) of matrices U ν . This is the most important structural property of U ν to be preserved. The second important property is the so called reduction group, see Section 6.
For any pair of solutions of (1.1) one can “compute” D := ΨΨ -1 . The crucial point is, however, to express D solely by the wave function Ψ because onl
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