Division of differential operators, intertwine relations and darboux transformations

Probing the Structure of Quantum Mechanics

2002

Links of factorization theory, supersymmetry and Darboux transformations as isospectral deformations of difference operators are considered in the context of soliton theory. The dressing chain equations for factorizing operators of a spectral problem are derived. The chain equations itselves yield nonlinear systems which closure generates solutions of the equations as well as of the nonlinear system if both operators of the correspondent Hirota bilinearization are covariant with respect to Darboux transformation which hence defines a symmetry of the nonlinear system as well as of these closed chains. Examples of Hirota and Nahm equations are specified. 1Introduction.The growing interest to discrete models appeal to necessity to widen classes of symmetry structures of the correspondent nonlinear problems [1]. Such theories as conformal field result in set of equations to be investigated [2]. [13]. The approach cover soliton, rational, finite-gap and other new solutions inside universal scheme, reducing the problem to a solution of closed sets of nonlinear ordinary equations with bi-Hamiltonian structure [13].In this paper we reformulate Darboux covariance theorem from [5] introducing a kind of the difference Bell polynomials. Such polynomials have natural correspondence to the differential (generalized) Bell polynomials in its nonabelian version [7], [8]. Its usage shortens the transformation formulas and helps to apply the theory in complicated cases of joint covariance of U-V pairs [9], [6], [16] on a way of the dressing chain equations derivation.The example of Hirota equation [14], connected with known applications, is studied in [5]. Here we derive the dressing chain equations and study simplest solutions of it. In the last section we propose the lattice Lax pair for Nahm equations 1

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confidence: 99%

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confidence: 99%

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Dressing Chain Equations Associated With Difference Soliton Systems

Probing the Structure of Quantum Mechanics

2002

“…where B n are differential Bell (Faa de Bruno [3]) polynomials [2]. The relation (4) generalizes so-called Miura map and became the identity when σ = φ ′ φ −1 , φ is a solution of the equation (2).…”

Section: Investigations Of General Darboux Transformation (Dt) Theorymentioning

confidence: 99%

“…The operator ∂ = ∂/∂x may be considered as a general differentiation as in [2]. The transformed potentialũ…”

Section: Joint Covariance Conditions For General Zs Equationsmentioning

confidence: 99%

Covariant Forms of Lax One-Field Operators: From Abelian to Noncommutative

Bilinear Integrable Systems: From Classical to Quantum, Continuous to Discrete

Abstract. Polynomials in differentiation operators are considered. Joint covariance with respect to Darboux transformations of a pair of such polynomials (Lax pair) as a function of one-field is studied. Methodically, the transforms of the coefficients are equalized to Frechèt differential (first term of the Taylor series on prolonged space) to establish the operator forms. In the commutative (Abelian) case, as it was recently proved for the KP-KdV Lax operators, it results in binary Bell (Faa de Bruno) differential polynomials having natural bilinear (Hirota) representation. Now next example of generalized Boussinesq equation with variable coefficients is studied, the dressing chain equations for the pair are derived. For a pair of generalized Zakharov-Shabat problems a set of integrable (non-commutative) potentials and hence nonlinear equations are constructed altogether with explicit dressing formulas. Some non-Abelian special functions are introduced.

Necessary Covariance Conditions for a One-Field Lax Pair

2005

Theor Math Phys

We study the covariance with respect to Darboux transformations of polynomial differential and difference operators with coefficients given by functions of one basic field. In the scalar (Abelian) case, the functional dependence is established by equating the Frechèt differential (the first term of the Taylor series on the prolonged space) to the Darboux transform; a Lax pair for the Boussinesq equation is considered. For a pair of generalized Zakharov-Shabat problems (with differential and shift operators) with operator coefficients, we construct a set of integrable nonlinear equations together with explicit dressing formulas. Non-Abelian special functions are fixed as the fields of the covariant pairs. We introduce a difference Lax pair, a combined gauge-Darboux transformation, and solutions of the Nahm equations.