Recursion in curve geometry
type: Item
instance of: scholarly article
title: Recursion in curve geometry
zbMATH Open document ID: 0930.35151
published in: The New York Journal of Mathematics
publication date: 1999-06-20T00:00:00Z
full work available at URL: https://eudml.org/doc/120108
full work available at URL: 5-3.html
review text: 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).
Mathematics Subject Classification ID: 35Q51
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 37K25
zbMATH DE Number: 1305702
zbMATH Keywords: filament model recursion
zbMATH Keywords: soliton curve
zbMATH Keywords: Frénet frame
zbMATH Keywords: nonlinear Schrödinger equation
zbMATH Keywords: modified Korteweg-de Vries equation
author: Joel C. Langer
reviewed by: Jan Chrastina
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1305702