The KP equation with self-consistent sources (KPESCS) is treated in the framework of the constrained KP equation. This offers a natural way to obtain the Lax representation for the KPESCS. Based on the conjugate Lax pairs, we construct the generalized binary Darboux transformation with arbitrary functions in time t for the KPESCS which, in contrast with the binary Darboux transformation of the ...

This paper studies the reconstruction of unknown curved surfaces through finger tracking. A patch can be generated from tactile data points along three concurrent surface curves under the Darboux frame estimated at the curve intersection point. Surface fitting while minimizing the total (absolute) Gaussian curvature effectively prevents unnecessary folds otherwise expected to result from the us...

We consider Bézier-like formulas with weights in quaternion and geometric (Clifford) algebra for parametrizing rational curves and surfaces. The simplest non-trivial quaternionic case of bilinear formulas for surface patches is studied in detail. Such formulas reproduce well known biquadratic parametrizations of principal Dupin cyclide patches, and are characterized in general as special Darbou...

The main scope of this paper is to examine the smoke ring (or vortex filament) equation which can be viewed as a dynamical system on space curve in E³. differential geometric properties soliton surface accociated with Nonlinear Schrödinger (NLS) equation, called NLS or Hasimoto surface, are investigated by using Darboux frame. Moreover, Gaussian and mean curvature found terms curvatures k_{n}, ...

We construct the Darboux transformation with Dihedral reduction group for the 2–dimensional generalisation of the periodic Volterra lattice. The resulting Bäcklund transformation can be viewed as a nonevolutionary integrable differential difference equation. We also find its generalised symmetry and the Lax representation for this symmetry. Using formal diagonalisation of the Darboux matrix we ...

Integrable discretisations for a class of coupled (super) nonlinear Schrödinger (NLS) type of equations are presented. The class corresponds to a Lax operator with entries in a Grassmann algebra. Elementary Darboux transformations are constructed. As a result, Grassmann generalisations of the Toda lattice and the NLS dressing chain are obtained. The compatibility (Bianchi commutativity) of thes...

The quantum oscillator thermodynamic actions are the conjugate intensive parameters for the frequency in any frequency changing process. These oscillator actions fulfill simple Riccati equations. Interesting Darboux transformations of the fundamental Planck and pure vacuum actions are discussed here in some detail. It is shown that the one-parameter “Darboux-Transformed-Thermodynamics” refers t...

We introduce several techniques which allow to simplify the expression of the cofactor of Darboux polynomials of polynomial differential systems in Rn. We apply these techniques to some well-known systems when n = 2, 3, 4. We also propose a general method for computing Darboux polynomials in the plane. As an application we prove that a family of potential systems, that includes the van der Pol ...

We have derived a necessary and sufficient condition for non-null normal spacelike curve lying in or timelike surface M ⊂ E13, so that the becomes K-type slant helix with K ∈ {1,2,3}. used Darboux frame to define conditions. An example is given 1-type having binormal.

The task of recognizing an algebraic surface from a single apparent contour can be reduced to the recovering homogeneous equation in four variables its discriminant. In this paper, we use fact that Darboux cyclides have singularity along absolute conic order recognize them up Euclidean similarity transformations.