On Bianchi and Bäcklund Transformations of Two-Dimensional Surfaces in E4

In this paper we discuss the existence and properties of the Bianchi transformations for pseudospherical surfaces in E ⁴. The results of the paper show that the theory of Bianchi transformations in the discussed case is essentially different from the well-known case of pseudospherical surfaces in E ³ (in general n-manifolds of constant and negative curvature in E ^{2 n − 1}). This revised version was published online in July 2006 with corrections to the Cover Date.     

Vortex filament motion in terms of Jacobi theta functions *

The paper summarizes a solution generation technique for the Localized-Induction-Approximation eqs. based on the approach of soliton surfaces. The technique is then applied to compute a four-real-parameter family of localized excitations of an arbitrary vortex filament motion of Kida class. The resulting formulae involve Jacobi theta functions.     

Bäcklund transformations for hyperbolic surfaces inE 3 via Weingarten congruences

An investigation of the so-called Weingarten congruences inE ³ yields a system of partial differential equations (describing hyperbolic surfaces inE ³) and also its Bäcklund transformation.     

N-solitons on a vortex filament

We describe a purely algebraic algorithm for finding, in the localized induction approach, three-dimensional vortex motion corresponding to the N-soliton solution of the nonlinear Schrödinger equation. The method used relies essentially up-on the geometric approach to soliton systems and consists in the construction of the corresponding soliton surfaces. For N = 0 and N = 1 we recover the straight line and Hasimoto vortex, respectively, while for N = 2 the algorithm gives a new vortex motion, of higher complexity, which is, presumably, an analytic expression for some scattering phenomena along vortices recently discovered by Browand and Hopfinger.