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We derive a permutability theorem for the Christoffel, Goursat and Darboux transformations of isothermic surfaces. As a consequence we obtain a simple proof of a relation between Darboux pairs of minimal surfaces in Euclidean space, curved flats in the 2-sphere and flat fronts in hyperbolic space.
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Formulas relating Poincare-Steklov operators for Schroedinger equations related by Darboux-Moutard transformations are derived. They can be used for testing algorithms of reconstruction of the potential from measurements at the boundary.
Singular Darboux transformations, in contrast to the conventional ones, have a singular matrix as a coefficient before the derivative. We incorporated such transformations into a chain of conventional transformations and presented determinant formulas for the resulting action of the chain. A determinant representation of the Kohlhoff-von Geramb solution to the Marchenko equation is given.
Deformations of the structure constants for a class of associative noncommutative algebras generated by Deformation Driving Algebras (DDAs) are defined and studied. These deformations are governed by the Central System (CS). Such a CS is studied for the case of DDA being the algebra of shifts. Concrete examples of deformations for the three-dimensional algebra governed by discrete and mixed continuous-discrete Boussinesq (BSQ) and WDVV equations are presented. It is shown that the theory of the Darboux transformations, at least for the BSQ case, is completely incorporated into the proposed scheme of deformations.
We consider stochastic UL and LU block factorizations of the one-step transition probability matrix for a discrete-time quasi-birth-and-death process, namely a stochastic block tridiagonal matrix. The simpler case of random walks with only nearest neighbors transitions gives a unique LU factorization and a one-parameter family of factorizations in the UL case. The block structure considered here yields many more possible factorizations resulting in a much enlarged class of potential applications. By reversing the order of the factors (also known as a Darboux transformation) we get new families of quasi-birth-and-death processes where it is possible to identify the matrix-valued spectral measures in terms of a Geronimus (UL) or a Christoffel (LU) transformation of the original one. We apply our results to one example going with matrix-valued Jacobi polynomials arising in group representation theory. We also give urn models for some particular cases.
Two binary (integral type) Darboux transformations for the KdV hierarchy with self-consistent sources are proposed. In contrast with the Darboux transformation for the KdV hierarchy, one of the two binary Darboux transformations provides non auto-B{a}cklund transformation between two n-th KdV equations with self-consistent sources with different degrees. The formula for the m-times repeated binary Darboux transformations are presented. This enables us to construct the N-soliton solution for the KdV hierarchy with self-consistent sources.
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