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A Dressing Method in Mathematical PhysicsImportant links

A Dressing Method in Mathematical Physics: Important links In this chapter we sketch some important links between ideas of the dress- ing Darboux transformation (DT), Bac ¨ klund transformation (BT), etc. with related mathematical constructions. Firstly, it is the Hirota representation which originally produced many of the known families of multisoliton solu- tions, and these have often led to a disclosure of the underlying Lax systems and infinite sets of conserved quantities [209, 385]. In Sect. 7.1 we demon- strate a systematic derivation of the bilinear BTs from the so-called Y-systems which are formulated in terms of the binary Bell polynomials. Taking as the example equations with the “sech ” soliton solutions, we illustrate how to obtain the binary BTs for different weights of the Y-polynomials. In Sect. 7.2 we represent the Darboux covariant Lax pairs in terms of the Y-systems. In Sect. 7.3 we explain how to construct BTs from the explicit dressing formu- las and, using the Noether theorem, how to derive discrete and continuous conservation laws. Next, in Sect. 7.4 the main formulas of the dressing theory are retrieved within the Weiss–Tabor–Carnevale procedure [449] of Painlev´ e analysis for partial differential equations (PDEs). In addition, we comment on a historical point connected with the http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

A Dressing Method in Mathematical PhysicsImportant links

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Publisher
Springer Netherlands
Copyright
© Springer 2007
ISBN
978-1-4020-6138-7
Pages
199–223
DOI
10.1007/1-4020-6140-4_7
Publisher site
See Chapter on Publisher Site

Abstract

In this chapter we sketch some important links between ideas of the dress- ing Darboux transformation (DT), Bac ¨ klund transformation (BT), etc. with related mathematical constructions. Firstly, it is the Hirota representation which originally produced many of the known families of multisoliton solu- tions, and these have often led to a disclosure of the underlying Lax systems and infinite sets of conserved quantities [209, 385]. In Sect. 7.1 we demon- strate a systematic derivation of the bilinear BTs from the so-called Y-systems which are formulated in terms of the binary Bell polynomials. Taking as the example equations with the “sech ” soliton solutions, we illustrate how to obtain the binary BTs for different weights of the Y-polynomials. In Sect. 7.2 we represent the Darboux covariant Lax pairs in terms of the Y-systems. In Sect. 7.3 we explain how to construct BTs from the explicit dressing formu- las and, using the Noether theorem, how to derive discrete and continuous conservation laws. Next, in Sect. 7.4 the main formulas of the dressing theory are retrieved within the Weiss–Tabor–Carnevale procedure [449] of Painlev´ e analysis for partial differential equations (PDEs). In addition, we comment on a historical point connected with the

Published: Jan 1, 2007

Keywords: Boussinesq Equation; Darboux Transformation; Bell Polynomial; Hirota Method; Moutard Transformation

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