Recursion in curve geometry
The article provides a concise but admirable survey of recent achievements concerning the interplay between soliton theory and classical differential geometry of curves in three-dimensional Euclidean space. A special filament model (FM) recursion inspired by a model of thin vortex tubes in an ideal fluid serves for a starting point: We introduce the recurrence \[ {d\gamma\over ds}\times X_n= {dX_{n-1}\over ds}\quad (n= 1,2,\dots),\quad X_0= -{d\gamma\over ds} \] for a given curve \(\gamma= \gamma(s)\) parametrized by the length \(s\). Then the general solution conveniently represented by a formal series \(X= \sum \lambda^n X_n\) is of the norm \(\|X\|= p(\lambda)\) independent of \(s\), and the normalization \(Y= X/\sqrt{p(\lambda)}\) can be expressed by differential operators acting on \(\gamma\). The \(n\)th soliton class of curves \(\gamma\) defined by \(X_n= 0\) can be alternatively characterized by variational problems. The standard Frénet frames of solitons are expressible by quadrature. The alternative natural Frénet frames are introduced with a parameter \(\sigma\) to provide a later link to the common solitons. The curve dynamics \(\partial\gamma/\partial t= X_n(\gamma)\) is of the Hamiltonian kind, there is an infinite number of conservation laws. The relationships to the nonlinear Schrödinger (NLS), modified Korteweg-de Vries (and sine-Gordon) equations are derived as a corollary to the variation of natural curvatures. At last, all the results are reformulated in terms of the Lie group SO(3). This permits to interpret the parameter \(\sigma\) of the natural Frénet equations as a spectral parameter and to identify the primary FM recursion with the common NLS recursion hierarchy (briefly derived from the zero curvature condition).
scientific article; zbMATH DE number 1305702