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From elementary to twofold elementary Darboux transformation In this chapter we extend the results of Chap. 2 related to the classical Dar- boux transformation (DT), by means of more detailed analysis of algebraic aspects of general theory. Indeed, already in the pioneering paper by Matveev [314] it was shown that the DT represents a universal algebraic operation. We start from the intertwining relations (Sect. 1.1) and formulate in Sect. 3.1 a general deﬁnition of the DT, as well as its connection with gauge transforma- tions. We introduce a concept of the elementary DT (eDT) [278] which will play a similar role for constructing particular solutions of nonlinear equations as the classical DT does (for a comprehensive study of the method see [433]). In Sect. 3.2 we begin the development of a purely algebraic construction of a matrix DT on the basis of two projectors [289]. The extension of the eDT covariance based on the existence of idempotents and skew ﬁelds in an as- sociative diﬀerential ring is discussed in Sect. 3.3 using an example of three basic projectors [267]. We stress that the twofold DT widely used as a dress- ing tool represents a sequence of two eDTs deﬁned
Published: Jan 1, 2007
Keywords: Spectral Parameter; Darboux Transformation; Conjugate Problem; Seed Solution; Inverse Element
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