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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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