The structure properties of multidimensional Delsarte-Darboux transmutation operators in parametric functional spaces is studied by means of differential-geometric and topological tools. It is shown that kernels of the corresponding integral operator expressions depend on the topological structure of related homological cycles in the coordinate space. As a natural realization of the constructio...

In this paper, the Dual Darboux rotation axis for timelike dual space curve in the semi-dual space D 1 is decomposed in two simultaneous rotation. The axis of these simultaneous rotations are joined by a simple mechanism. This study is original corresponding in the semi-dual space D 1 of the article entitled On the Dual Darboux Rotation Axis of the Dual Space Curve [6]. Mathematics Subject Clas...

Note that Blaschke and his school only considered surfaces in 3-space and ignored the higher co-dimension case. In Sn this problem is still meaningful. It is known that the construction of Darboux pair of isothermic surfaces as well as dual Willmore surfaces has a generalization to Sn (see [4, 9] and [7]), and they still constitute solutions to the generalized Blaschke’s problem. Here we will s...

Abstract We consider a non-Hermitian matrix orthogonality on contour in the complex plane. Given diagonalizable and rational valued weight, we show that Christoffel–Darboux (CD) kernel, which is built terms of orthogonal polynomials, equivalent to scalar reproducing kernel meromorphic functions Riemann surface. If this surface has genus $0$, then CD polynomials Interestingly, not necessarily ke...

In this paper, we focus on a developable surface tangent to timelike along curve in Minkowski 3-space, which is called the osculating of curve. The ruling parallel Darboux vector field. main goal paper classify singularities surface. To end, two new invariants curves are defined characterize these singularities. Meanwhile, also research singular properties surfaces near their lightlike points. ...

The Marden theorem of geometry of polynomials and the great Poncelet theorem from projective geometry of conics by their classical beauty occupy very special places. Our main aim is to present a strong and unexpected relationship between the two theorems. We establish a dynamical equivalence between the full Marden theorem and the Poncelet-Darboux theorem. By introducing a class of isofocal def...

In this article, the Darboux transformation for the non-isospectral AKNS hierarchy is constructed. We show that the Darboux transformation for the non-isospectral AKNS hierarchy is not an auto-Bäcklund transformation, because the integral constants of the hierarchy will be changed after the transformation. The transform rule of the integral constants will be also derived. By this means, the sol...

A noncommutative version of the KP equation and two families of its solutions expressed as quasideterminants are discussed. The origin of these solutions is explained by means of Darboux and binary Darboux transformations. Additionally, it is shown that these solutions may also be verified directly. This approach is reminiscent of the wronskian technique used for the Hirota bilinear form of the...

In this paper, the third kind of Darboux transformation of generalized Broer–Kaup equations is derived from the corresponding spectral problem. By virtue of this Darboux transformation, new 2N -soliton solutions with parameters of the generalized Broer–Kaup equations are obtained. Although 2N is an even number, it is graphically shown that in the cases of N= 1 and N = 2 the obtained 2N -soliton...