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A Dressing Method in Mathematical PhysicsFactorization and classical Darboux transformations

A Dressing Method in Mathematical Physics: Factorization and classical Darboux transformations Factorization and classical Darboux transformations In this chapter we describe the algebraical factorization-based method to dress solutions of (1+1)-dimensional equations. We also show how the Darboux transformation (DT) theory appears in this framework. First, in Sect. 2.1, we introduce the non-Abelian Bell polynomials and then generalize them in Sect. 2.2 to formulate in Sect. 2.3 a problem of fac- torization of a polynomial differential operator in the form of division by a monomial from the right and from the left. The relation between the factor- ization rules and the classical Darboux theorem [102] generalized in [314] is described in Sect. 2.4: the formalism produces a compact form of the DT for non-Abelian coefficients of linear operators, polynomial in a differentiation on a ring. Section 2.5 is devoted to a representation of the iterated DTs in terms of quasideterminants. As a highly nontrivial example of the iterated DT formalism, we describe positon solutions of the Korteweg–de Vries (KdV) equation discovered by Matveev [318, 319]. The growing interest in discrete models appeals to wider classes of sym- metry structures of the corresponding nonlinear problems [149, 196, 255, 256, 339]. Very recently a suitable basis for new searches in the field of http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

A Dressing Method in Mathematical PhysicsFactorization and classical Darboux transformations

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Publisher
Springer Netherlands
Copyright
© Springer 2007
ISBN
978-1-4020-6138-7
Pages
31 –65
DOI
10.1007/1-4020-6140-4_2
Publisher site
See Chapter on Publisher Site

Abstract

Factorization and classical Darboux transformations In this chapter we describe the algebraical factorization-based method to dress solutions of (1+1)-dimensional equations. We also show how the Darboux transformation (DT) theory appears in this framework. First, in Sect. 2.1, we introduce the non-Abelian Bell polynomials and then generalize them in Sect. 2.2 to formulate in Sect. 2.3 a problem of fac- torization of a polynomial differential operator in the form of division by a monomial from the right and from the left. The relation between the factor- ization rules and the classical Darboux theorem [102] generalized in [314] is described in Sect. 2.4: the formalism produces a compact form of the DT for non-Abelian coefficients of linear operators, polynomial in a differentiation on a ring. Section 2.5 is devoted to a representation of the iterated DTs in terms of quasideterminants. As a highly nontrivial example of the iterated DT formalism, we describe positon solutions of the Korteweg–de Vries (KdV) equation discovered by Matveev [318, 319]. The growing interest in discrete models appeals to wider classes of sym- metry structures of the corresponding nonlinear problems [149, 196, 255, 256, 339]. Very recently a suitable basis for new searches in the field of

Published: Jan 1, 2007

Keywords: Spectral Problem; Darboux Transformation; Spectral Equation; Bell Polynomial; Necessity Condition

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