相关论文: Motion of Curves on Two Dimensional Surfaces and S…

Some aspects of the relation between differential geometry of curves and surfaces and multidimensional soliton equations is discussed. The connection between multidimensional soliton equations and Self-dual Yang-Mills equation is studied.

A connection between differential geometry and soliton equations is discussed

The subject of moving curves (and surfaces) in three dimensional space (3-D) is a fascinating topic not only because it represents typical nonlinear dynamical systems in classical mechanics, but also finds important applications in a…

Lamb has identified a certain class of moving space curves with soliton equations. We show that there are two other classes of curve evolution that may be so identified. Hence three distinct classes of curve evolution are associated with a…

Motion of curves and surfaces in $\R^3$ lead to nonlinear evolution equations which are often integrable. They are also intimately connected to the dynamics of spin chains in the continuum limit and integrable soliton systems through…

We apply our recent formalism establishing new connections between the geometry of moving space curves and soliton equations, to the nonlinear Schr\"{o}dinger equation (NLS). We show that any given solution of the NLS gets associated with…

It is shown that a class of important integrable nonlinear evolution equations in (2+1) dimensions can be associated with the motion of space curves endowed with an extra spatial variable or equivalently, moving surfaces. Geometrical…

Starting with the general description of a moving curve, we have recently presented a unified formalism to show that three distinct space curve evolutions can be identified with a given integrable equation. Applying this to the nonlinear…

Some aspects of the connection between differential geometry and multidimensional soliton equations are discussed.

For a polygon in Euclidean space we consider a transformation T which is obtained by applying the midpoints polygon construction twice and using an index shift. For a closed polygon this is a curve shortening process. A polygon is called…

In this paper we study similarity measures for moving curves which can, for example, model changing coastlines or retreating glacier termini. Points on a moving curve have two parameters, namely the position along the curve as well as time.…

A new kinematic condition for soliton motions of an $n$-dimensional continuum in $R^{n+m}$, independent of the underlying physics, is proven. The condition and its consequences for different cases are demonstrated. A soliton in a 1D string…

In the half-space model of the hyperbolic three space with the hyperbolic metric, this same space can be seen as the Lie group, hence, a translation surface is a surface that is given by the product of two curves $\alpha$ and $\beta$ in…

We describe the curves of constant (geodesic) curvature and torsion in the three-dimensional round sphere. These curves are the trajectory of a point whose motion is the superposition of two circular motions in orthogonal planes. The global…

By considering a spatial curve in a Euclidean space, we use its components, together with attaining a cyclic matrix, to show that this matrix is homothetic too and is in correspondence with a homothetic motion. Furthermore, if the curve…

We investigate the vertex curve, that is the set of points in the hyperbolic region of a smooth surface in real 3-space at which there is a circle in the tangent plane having at least 5-point contact with the surface. The vertex curve is…

Some aspects of the multidimensional soliton geometry are considered.

We show that a curve is a soliton solution to the curve shortening flow if and only if its geodesic curvature can be written as the inner product between its tangent vector field and a fixed vector of the 3-dimensional Minkowski space. We…

The connection between multidimensional soliton equations and three-dimensional Riemann space is discussed.

In this paper, we construct rotating frames for curves, including plane curves, space curves and curves on surfaces. Hence, the behaviour of an arbitrary moving point on a curve can be seen as the composite of linear motion and rotation.…