Soliton Surfaces Research Articles

Isothermic surfaces in E 3 as soliton surfaces

We show that the theory of isothermic surfaces in E 3 — one of the oldest branches of differential geometry — can be reformulated within the modern theory of completely integrable (soliton) systems. This enables one to study the geometry of isothermic surfaces in E 3 by means of powerful spectral methods available in the soliton theory. Also the associated non-linear system is interesting in itself since it displays some unconventional soliton features and, physically, could be applied in the theory of infinitesimal deformations of membranes.

An effective method to compute N-fold Darboux matrix and N-soliton surfaces

Fairly simple formulas for an N-fold Darboux matrix and an N-soliton addition to an arbitrary soliton surface embedded in E3 are proven. They depend directly on the background wave function and are evidently symmetric with respect to N added solitons. As an example, an explicit expression for the interaction of N solitons on a single vortex filament is presented. Moreover, there is given a simplified version of the N-soliton formulas of Neugebauer and Meinel.

Vortex filament motion in terms of Jacobi theta functions**Work supported in part by Polish Ministry of Science and Higher Education, Research Problem CPBP 01.03

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.

Soliton equations and surfaces of constant curvature

Soliton equations and surfaces of constant curvature

THE STATIC AXIALLY SYMMETRIC SELF-DUAL YANG-MILLS FIELDS AND SURFACES OF NEGATIVE CURVATURE

An explicit geometric picture about the complete integrability of the static axially symmetric self-dual Yang-Mills equation and the gravitational Ernst equation is presented. The corresponding soliton surfaces in adjoint space (3-dimensional Minkowski space) has negative variable curvature. The Riccati equation is also given, so that the integrability of the Bäcklund transformation gets the confirmation.

Self-Dual Yang-Mills Fields in Static Axially Symmetric Case and Related Topics

In this article we will present an explicit geometric picture about the complete integrability of the static axially symmetric SDYM equation and the gravitational Ernst equation, interpret the correspondence between their Bäcklund transformation formulae and the transformations from one focal surface of Weingarten congruence to the other, and give the matrix Riccati equation so that the integrability of the B.T. will be proved. It is shown that for the axially symmetric SDYM equation and gravitational Ernst equation the adjoint space of the group (SL(2r)) is a 3-dimensional Minkowski space, and the corresponding soliton surfaces have negative variable curvature. After introducing the generator R we can explain the B.T. as the rotation around the common tangent between two surfaces of solitons. Using Riccati equation we will confirm in this paper the integrability of B.T. and prove that the B.T. is strong, i.e., the new and old solutions satisfy equations of motion separately. Some related topics are also discussed.

Soliton surfaces

It is shown that two arbitrary soliton systems withsw 2 (su 1,1) spectral problems are connected by a gauge transformation plus a change of independent variables, if and only if the corresponding pairs of fundamental tensors are identical. There are two advantages of this criterion: 1) no need to calculate a gauge matrix, 2) an appropriate change of independent variables and the corresponding nonlinear transformation of soliton fields follow directly from geometry. As an illustration it is shown that the « new integrable nonlinear evolution equations » possess the old (AKNS) soliton surfaces. Moreover, the final formulation of the approach of soliton surfaces is given.

Soliton surfaces and their applications

Soliton surfaces and their applications

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.

Soliton surfaces

Two applications of the concept of soliton surfaces are discussed. Firstly, soliton surfaces can serve as a territory of unification of four types of solvable nonlinearities: 1) soliton, 2) strings, 3) spins and 4) chiral models. It is conjectured that models 2), 3) and 4) associated with a given soliton system are gauge equivalent to this soliton system. Secondly, an explicit construction of the soliton surface associated with a given soliton solution gives simultaneously the corresponding solutions to models 2), 3) and 4). Using the Hilbert-Riemann problem technique a construction of N-soliton surfaces is described. Examples including new soliton systems are given.