Generalized Bäcklund–Darboux Transformation: Spectral Propertiesand Nonlinear Equations

By Alexander L. Sakhnovich. 2001

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The Bäcklund–Darboux transformations (BDTs) are named after the pioneering discoveries by Bäcklund [6] and Darboux [14, 15] in differential geometry. The problems under consideration were related to the sineGordon equation ∂2u/∂t ∂x = sinu. In particular, a transformation producing a new solution of the sine-Gordon equation from a given solution was introduced. After being an active research area for some time after the turn of the (previous) century, Bäcklund transformations were mostly forgotten until they came back with a vengeance in the 1970s following the discovery of complete integrability of nonlinear PDEs such as the sine-Gordon equation, the Korteweg–de Vries equation, and the nonlinear Schrödinger equation, of their soliton solutions and of the inverse scattering method. See [2, 13, 30, 31, 45] for the achievements of the last few decades. The name Bäcklund transformation refers sometimes to any system of functional equations involving two functions u and ũ and their derivatives, such that if u satisfies one given PDE then ũ will satisfy another given PDE (often coinciding with the first PDE) and conversely (see, e.g., [2]). The functional equations typically contain a constant playing the role of a “spectral parameter.” On the other hand a similar transformation is essential in the spectral theory of the ODE: Crum–Krein–Agranovich–Marchenko formulas, insertion of additional bound states, trace formulas, commutation methods, and bispectrality (see [4, 17–19, 21, 31]). An eigenvalue λ0 can be

Subfields: Spectroscopy, Statistical and Nonlinear Physics

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