Bäcklund transformations of (2+1)-dimensional integrable systems

Abstract

(the theory of Egorov curvilinear orthogonal systems of coordinates in R3), and some other cases. They also derived nonlinear superposition formulas for all these cases. In the modern theory of (1 + I)-dimensional nonlinear integrable systems of partial differential equations, the B~icklund transformations (see [16, 17]) and the corresponding nonlinear superposition formulas have been obtained for most such systems.A theory of Bacldund transformations is also .available for (2 + l)-dimensional nonlinear integrable equations (i.e., equations with 2 "space" and 1 "time" independent variables). Such problems (described by equations or systems of equations with three independent variables) also were intensively studied in the classical differential geometry at the turn of the century: the theory of arbitrary curvilinear orthogonal systems of coordinates in R3 (and also in En, which is a (2 + 1)-dimensional 38

Three–dimensional integrable lattices in Euclidean spaces: conjugacy and orthogonality

“…In this connection, we point out that it was Bianchi who derived a Bäcklund transformation for triply orthogonal systems of surfaces which was based on the Ribaucour transformation. Remarkably, he demonstrated that, for any choice of the parameters u i , the position vector r of the seed orthogonal coordinate system and its Bäcklund transforms r 1 , r 2 , r 12 = r 21 lie on a circle (Bianchi 1923(Bianchi -1927Ganzha & Tsarev 1996). Thus, iterative application of Bianchi's transformation generates cyclic and hence orthogonal lattices (cf.…”

Vectorial Ribaucour transformations for the Lamé equations

“…Later in [1], see also [2], it was done in three dimensional case and in [9] one can find the extension to any dimension. Recently, in [10] three Ribaucour transformations were iterated in 3-dimensional space to get some results related with permutability. The permutability theorem for the scalar fundamental was established in [11].…”

Generating quadrilateral and circular lattices in KP theory

“…For example, for j = i, the use of (6), (20) and (15) leads to the desired result. The i-th derivative must be treated with more care, in fact, we have to start with the alternative expressions…”

“…For j = i one just take the derivative and then uses ( 9), ( 6), (20) and (15). The j = i case requires the use of the following expression for C (i) ( it derives from (20))…”

“…The iteration of two Ribaucour transformation was performed by Demoulin [10] finding the famous circularity property of these transformations. Recently, the vectorial Ribaucour transformation [28] has been presented and shown to be equivalent to the iteration of the standard ones, see also [19,20].…”