For the n-dimensional integrable system with a twisted so(p, q) reduction, Darboux transformations given by Darboux matrices of degree 2 are constructed explicitly. These Darboux transformations are applied to the local isometric immersion of space forms with flat normal bundle and linearly independent curvature normals to give the explicit expression of the position vector. Some examples are g...

Darboux transformations in one variable form the basis for the factorization methods and have numerous applications to geometry, nonlinear equations and SUSY quantum mechanics. In spite of this wide range of applications the theory of Darboux transformations in two variables and its elegant relationship to analytic complex functions has not been recognized in the literature. To close this gap w...

For operators of many different kinds it has been proved that (generalized) Darboux transformations can be built using so called Wronskian formulae. Such Darboux transformations are not invertible in the sense that the corresponding mappings of the operator kernels are not invertible. The only known invertible ones were Laplace transformations (and their compositions), which are special cases o...

In this paper, we define X-slant helix in Euclidean 3-space and we obtain helix, slant helix, clad and g-clad helix as special case of the X-slant helix. Then we study Darboux, tangential darboux developable surfaces and their singular points. Especially we show that the striction lines of these surfaces are singular locus of the surfaces.

An essential ingredient of the conventional supersymmetric quantum mechanics (for reviews see [1]) is the well known Darboux transformation [2] for the stationary Schrödinger equation. This transformation permits us to construct new exactly solvable stationary potentials from known ones. Similar constructions may be developed for the time-dependent Schrödinger equation [3]. Our approach to Darb...

where Aj = −d/dx + fj(x) are first order differential operators. A Darboux chain is called periodic if Lj+r = Lj for some r and for all j = 1, 2, . . . . Number r is called the period of a Darboux chain. The operator L+ α2 appears to be the harmonic oscillator in the particular case r = 1. Periodic Darboux chains lead to integrable systems of differential equations for functions fj, which are e...

For a generalized super KdV equation, three Darboux transformations and the corresponding Bäcklund transformations are constructed. The compatibility of these Darboux transformations leads to three discrete systems and their Lax representations. The reduction of one of the Bäcklund–Darboux transformations and the corresponding discrete system are considered for Kupershmidt’s super KdV equation....

In this paper we study the Darboux transformations of planar vector fields of Schrödinger type. Using the isogaloisian property of Darboux transformation we prove the “invariance” of the objects of the “Darboux theory of integrability”. In particular, we also show how the shape invariance property of the potential is important in order to preserve the structure of the transformed vector field. ...