In Euclidean 3-space, the Frenet and Darboux instantaneous rotation vectors for a curve (c) on the surface which the parameter curves are perpendicular to each other are known. Let us consider an arbitrary curve (c) and the parameter curves (Cl) , (cz) passing through a point P on surface. If the Darboux instantaneous rotation vectors of these curves are shown by W, WI and Wz respectively, then...

The fact that the Darboux frame is rotation–minimizing along lines of curvature of a smooth surface is invoked to construct rational surface patches whose boundary curves are lines of curvature. For given patch corner points and associated frames defining the surface normals and principal directions, the patch boundaries are constructed as quintic RRMF curves, i.e., spatial Pythagorean–hodograp...

A Darboux-transformed surface gravitational acceleration of the constant gravitational acceleration for a body endowed with an atmospheric layer is shown to turn the atmospheric free fall with quadratic resistance in the opposite motion, i.e., a free rising. Although the atmosphere of such a body may look completely normal, it is the time dependence of its gravitational field that produces this...

The question of the correct formulation of a Darboux type non-characteristic spatial problem for the wave equation is investigated. The correct solvability of the problem is proved in the Sobolev space for surfaces of the temporal type on which Darboux type boundary conditions are given. In the space of variables x1, x2, t let us consider the wave equation ƒu ≡ ∂ 2u ∂t2 − ∂ 2u ∂x1 − ∂ 2u ∂x2 2 ...

The multiplier spectral curve of a conformal torus f : T 2 → S in the 4– sphere is essentially [3] given by all Darboux transforms of f . In the particular case when the conformal immersion is a Hamiltonian stationary torus f : T 2 → R in Euclidean 4–space, the left normal N : M → S of f is harmonic, hence we can associate a second Riemann surface: the eigenline spectral curve of N , as defined...

We analyze Darboux transformations in very general settings for multidimensional linear partial differential operators. We consider all known types of Darboux transformations, and present a new type. We obtain a full classification of all operators that admit Wronskian type Darboux transformations of first order and a complete description of all possible first-order Darboux transformations. We ...

It is shown that for every Darboux function F there is a non-constant continuous function f such that F + f is still Darboux. It is shown to be consistent—the model used is iterated Sacks forcing—that for every Darboux function F there is a nowhere constant continuous function f such that F + f is still Darboux. This answers questions raised in [5] where it is shown that in various models of se...

A. M. Bruckner and J. B. Bruckner gave the definition of Darboux 53 functions and proved a theorem which is a local characterization of real-valued Darboux 58 functions. The purpose of this paper is to generalize this theorem. To this end, the definition of a function being Darboux S at a point is given which has a metric continuum as its range. Hence, the theorem that a function is Darboux 8 i...

In this study, we introduce a new type of surface curves called $D$-type curve. This curve is defined by the property that the unit Darboux vector $vec{W}_{0} $ of a surface curve $vec{r}(s)$ and unit surface normal $vec{n} $ along the curve $vec{r}(s)$ satisfy the condition $leftlangle vec{n} ,vec{W}_{0} rightrangle =text{constant}$. We point out that a $D$-type curve is a geodesic curve or an...

We construct Darboux transformations for the supersymmetric KP hierarchy. We show that the naive candidate does not work due to the pressure of the odd flows,but its composition does give rise to a meaningful Darboux transformation. We also consider the b inary Darboux transformation for the hierarchy. The iterations of both type of Darboux transformations are briefly discussed.